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| <div style="width: 100%; height: 75vh; overflow: auto;"> | | <div style="-moz-column-count:2; column-count:2;"> |
| {| class="wikitable sortable" style="margin: 0;"
| | ; Notation : [[12.1|12.1 Special Notation]]<br> |
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| | ; Properties : [[12.2|12.2 Differential Equations]]<br>[[12.3|12.3 Graphics]]<br>[[12.4|12.4 Power-Series Expansions]]<br>[[12.5|12.5 Integral Representations]]<br>[[12.6|12.6 Continued Fraction]]<br>[[12.7|12.7 Relations to Other Functions]]<br>[[12.8|12.8 Recurrence Relations and Derivatives]]<br>[[12.9|12.9 Asymptotic Expansions for Large Variable]]<br>[[12.10|12.10 Uniform Asymptotic Expansions for Large Parameter]]<br>[[12.11|12.11 Zeros]]<br>[[12.12|12.12 Integrals]]<br>[[12.13|12.13 Sums]]<br>[[12.14|12.14 The Function <math>\paraW@{a}{x}</math>]]<br>[[12.15|12.15 Generalized Parabolic Cylinder Functions]]<br> |
| ! scope="col" style="position: sticky; top: 0;" | DLMF
| | ; Applications : [[12.16|12.16 Mathematical Applications]]<br>[[12.17|12.17 Physical Applications]]<br> |
| ! scope="col" style="position: sticky; top: 0;" | Formula
| | ; Computation : [[12.18|12.18 Methods of Computation]]<br>[[12.19|12.19 Tables]]<br>[[12.20|12.20 Approximations]]<br>[[12.21|12.21 Software]]<br> |
| ! scope="col" style="position: sticky; top: 0;" | Constraints
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| ! scope="col" style="position: sticky; top: 0;" | Maple
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| ! scope="col" style="position: sticky; top: 0;" | Mathematica
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| ! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
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| ! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
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| ! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
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| ! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| | [https://dlmf.nist.gov/12.2.E2 12.2.E2] || [[Item:Q4087|<math>\deriv[2]{w}{z}-\left(\tfrac{1}{4}z^{2}+a\right)w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{w}{z}-\left(\tfrac{1}{4}z^{2}+a\right)w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [z$(2)])-((1)/(4)*(z)^(2)+ a)*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {z, 2}]-(Divide[1,4]*(z)^(2)+ a)*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.299038106+.4999999999*I
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| Test Values: {a = -3/2, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.299038106+1.000000000*I
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| Test Values: {a = -3/2, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.2990381056766582, 0.4999999999999999]
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| Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.299038105676658, 0.9999999999999999]
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| Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.2.E3 12.2.E3] || [[Item:Q4088|<math>\deriv[2]{w}{z}+\left(\tfrac{1}{4}z^{2}-a\right)w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{w}{z}+\left(\tfrac{1}{4}z^{2}-a\right)w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [z$(2)])+((1)/(4)*(z)^(2)- a)*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {z, 2}]+(Divide[1,4]*(z)^(2)- a)*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.299038106+1.000000000*I
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| Test Values: {a = -3/2, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.299038106+.4999999999*I
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| Test Values: {a = -3/2, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.299038105676658, 0.9999999999999999]
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| Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.2990381056766582, 0.4999999999999999]
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| Test Values: {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.2.E4 12.2.E4] || [[Item:Q4089|<math>\deriv[2]{w}{z}+\left(\nu+\tfrac{1}{2}-\tfrac{1}{4}z^{2}\right)w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{w}{z}+\left(\nu+\tfrac{1}{2}-\tfrac{1}{4}z^{2}\right)w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [z$(2)])+(nu +(1)/(2)-(1)/(4)*(z)^(2))*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {z, 2}]+(\[Nu]+Divide[1,2]-Divide[1,4]*(z)^(2))*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .9330127024+.8660254039*I
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| Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .9330127024+1.366025404*I
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| Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [296 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.9330127018922196, 0.8660254037844386]
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| Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.4330127018922191, 0.5000000000000001]
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| Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.2.E5 12.2.E5] || [[Item:Q4090|<math>\WhittakerparaD{\nu}@{z} = \paraU@{-\tfrac{1}{2}-\nu}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerparaD{\nu}@{z} = \paraU@{-\tfrac{1}{2}-\nu}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderD(nu, z) = CylinderU(-(1)/(2)- nu, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[\[Nu], z] == ParabolicCylinderD[- 1/2 -(-Divide[1,2]- \[Nu]), z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 70]
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| | [https://dlmf.nist.gov/12.2.E6 12.2.E6] || [[Item:Q4091|<math>\paraU@{a}{0} = \frac{\sqrt{\pi}}{2^{\frac{1}{2}a+\frac{1}{4}}\EulerGamma@{\frac{3}{4}+\frac{1}{2}a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{0} = \frac{\sqrt{\pi}}{2^{\frac{1}{2}a+\frac{1}{4}}\EulerGamma@{\frac{3}{4}+\frac{1}{2}a}}</syntaxhighlight> || <math>\realpart@@{(\frac{3}{4}+\frac{1}{2}a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderU(a, 0) = (sqrt(Pi))/((2)^((1)/(2)*a +(1)/(4))* GAMMA((3)/(4)+(1)/(2)*a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), 0] == Divide[Sqrt[Pi],(2)^(Divide[1,2]*a +Divide[1,4])* Gamma[Divide[3,4]+Divide[1,2]*a]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 4]
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| | [https://dlmf.nist.gov/12.2.E7 12.2.E7] || [[Item:Q4092|<math>\paraU'@{a}{0} = -\frac{\sqrt{\pi}}{2^{\frac{1}{2}a-\frac{1}{4}}\EulerGamma@{\frac{1}{4}+\frac{1}{2}a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU'@{a}{0} = -\frac{\sqrt{\pi}}{2^{\frac{1}{2}a-\frac{1}{4}}\EulerGamma@{\frac{1}{4}+\frac{1}{2}a}}</syntaxhighlight> || <math>\realpart@@{(\frac{1}{4}+\frac{1}{2}a)} > 0</math> || <syntaxhighlight lang=mathematica>subs( temp=0, diff( CylinderU(a, temp), temp$(1) ) ) = -(sqrt(Pi))/((2)^((1)/(2)*a -(1)/(4))* GAMMA((1)/(4)+(1)/(2)*a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[ParabolicCylinderD[- 1/2 -(a), temp], {temp, 1}]/.temp-> 0) == -Divide[Sqrt[Pi],(2)^(Divide[1,2]*a -Divide[1,4])* Gamma[Divide[1,4]+Divide[1,2]*a]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/12.2.E8 12.2.E8] || [[Item:Q4093|<math>\paraV@{a}{0} = \frac{\pi 2^{\frac{1}{2}a+\frac{1}{4}}}{\left(\EulerGamma@{\frac{3}{4}-\frac{1}{2}a}\right)^{2}\EulerGamma@{\frac{1}{4}+\frac{1}{2}a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV@{a}{0} = \frac{\pi 2^{\frac{1}{2}a+\frac{1}{4}}}{\left(\EulerGamma@{\frac{3}{4}-\frac{1}{2}a}\right)^{2}\EulerGamma@{\frac{1}{4}+\frac{1}{2}a}}</syntaxhighlight> || <math>\realpart@@{(\frac{3}{4}-\frac{1}{2}a)} > 0, \realpart@@{(\frac{1}{4}+\frac{1}{2}a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderV(a, 0) = (Pi*(2)^((1)/(2)*a +(1)/(4)))/((GAMMA((3)/(4)-(1)/(2)*a))^(2)* GAMMA((1)/(4)+(1)/(2)*a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, 0] + ParabolicCylinderD[-(a) - 1/2, -(0)]) == Divide[Pi*(2)^(Divide[1,2]*a +Divide[1,4]),(Gamma[Divide[3,4]-Divide[1,2]*a])^(2)* Gamma[Divide[1,4]+Divide[1,2]*a]]</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.7978845608028653, Times[0.7978845608028655, GAMMA[1.0]]]
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| Test Values: {Rule[a, 0.5]}</syntaxhighlight><br></div></div>
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| | [https://dlmf.nist.gov/12.2.E9 12.2.E9] || [[Item:Q4094|<math>\paraV'@{a}{0} = \frac{\pi 2^{\frac{1}{2}a+\frac{3}{4}}}{\left(\EulerGamma@{\frac{1}{4}-\frac{1}{2}a}\right)^{2}\EulerGamma@{\frac{3}{4}+\frac{1}{2}a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV'@{a}{0} = \frac{\pi 2^{\frac{1}{2}a+\frac{3}{4}}}{\left(\EulerGamma@{\frac{1}{4}-\frac{1}{2}a}\right)^{2}\EulerGamma@{\frac{3}{4}+\frac{1}{2}a}}</syntaxhighlight> || <math>\realpart@@{(\frac{1}{4}-\frac{1}{2}a)} > 0, \realpart@@{(\frac{3}{4}+\frac{1}{2}a)} > 0</math> || <syntaxhighlight lang=mathematica>subs( temp=0, diff( CylinderV(a, temp), temp$(1) ) ) = (Pi*(2)^((1)/(2)*a +(3)/(4)))/((GAMMA((1)/(4)-(1)/(2)*a))^(2)* GAMMA((3)/(4)+(1)/(2)*a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, temp] + ParabolicCylinderD[-(a) - 1/2, -(temp)]), {temp, 1}]/.temp-> 0) == Divide[Pi*(2)^(Divide[1,2]*a +Divide[3,4]),(Gamma[Divide[1,4]-Divide[1,2]*a])^(2)* Gamma[Divide[3,4]+Divide[1,2]*a]]</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -0.7978845608028653
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| Test Values: {Rule[a, -0.5]}</syntaxhighlight><br></div></div>
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| | [https://dlmf.nist.gov/12.2.E10 12.2.E10] || [[Item:Q4095|<math>\Wronskian@{\paraU@{a}{z},\paraV@{a}{z}} = \sqrt{2/\pi}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\paraU@{a}{z},\paraV@{a}{z}} = \sqrt{2/\pi}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(CylinderU(a, z))*diff(CylinderV(a, z), z)-diff(CylinderU(a, z), z)*(CylinderV(a, z)) = sqrt(2/Pi)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{ParabolicCylinderD[- 1/2 -(a), z], Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)])}, z] == Sqrt[2/Pi]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [14 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .708254234e-1-.722805450e-2*I
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| Test Values: {a = 3/2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .4257865765+.241883787*I
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| Test Values: {a = 3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.7978845608028654, Times[Complex[-3.533949646070574*^-17, -3.533949646070574*^-17], GAMMA[-1.0]]]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.7978845608028654, Times[Complex[0.0, -2.1203697876423444*^-16], GAMMA[-1.0]]]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.2.E11 12.2.E11] || [[Item:Q4096|<math>\Wronskian@{\paraU@{a}{z},\paraU@{a}{-z}} = \frac{\sqrt{2\pi}}{\EulerGamma@{\frac{1}{2}+a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\paraU@{a}{z},\paraU@{a}{-z}} = \frac{\sqrt{2\pi}}{\EulerGamma@{\frac{1}{2}+a}}</syntaxhighlight> || <math>\realpart@@{(\frac{1}{2}+a)} > 0</math> || <syntaxhighlight lang=mathematica>(CylinderU(a, z))*diff(CylinderU(a, - z), z)-diff(CylinderU(a, z), z)*(CylinderU(a, - z)) = (sqrt(2*Pi))/(GAMMA((1)/(2)+ a))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{ParabolicCylinderD[- 1/2 -(a), z], ParabolicCylinderD[- 1/2 -(a), - z]}, z] == Divide[Sqrt[2*Pi],Gamma[Divide[1,2]+ a]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
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| | [https://dlmf.nist.gov/12.2.E12 12.2.E12] || [[Item:Q4097|<math>\Wronskian@{\paraU@{a}{z},\paraU@{-a}{+ iz}} = - ie^{+ i\pi(\frac{1}{2}a+\frac{1}{4})}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\paraU@{a}{z},\paraU@{-a}{+ iz}} = - ie^{+ i\pi(\frac{1}{2}a+\frac{1}{4})}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(CylinderU(a, z))*diff(CylinderU(- a, + I*z), z)-diff(CylinderU(a, z), z)*(CylinderU(- a, + I*z)) = - I*exp(+ I*Pi*((1)/(2)*a +(1)/(4)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{ParabolicCylinderD[- 1/2 -(a), z], ParabolicCylinderD[- 1/2 -(- a), + I*z]}, z] == - I*Exp[+ I*Pi*(Divide[1,2]*a +Divide[1,4])]</syntaxhighlight> || Failure || Failure || Successful [Tested: 42] || Successful [Tested: 42]
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| | [https://dlmf.nist.gov/12.2.E12 12.2.E12] || [[Item:Q4097|<math>\Wronskian@{\paraU@{a}{z},\paraU@{-a}{- iz}} = + ie^{- i\pi(\frac{1}{2}a+\frac{1}{4})}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\paraU@{a}{z},\paraU@{-a}{- iz}} = + ie^{- i\pi(\frac{1}{2}a+\frac{1}{4})}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(CylinderU(a, z))*diff(CylinderU(- a, - I*z), z)-diff(CylinderU(a, z), z)*(CylinderU(- a, - I*z)) = + I*exp(- I*Pi*((1)/(2)*a +(1)/(4)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{ParabolicCylinderD[- 1/2 -(a), z], ParabolicCylinderD[- 1/2 -(- a), - I*z]}, z] == + I*Exp[- I*Pi*(Divide[1,2]*a +Divide[1,4])]</syntaxhighlight> || Failure || Failure || Successful [Tested: 42] || Successful [Tested: 42]
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| | [https://dlmf.nist.gov/12.2.E13 12.2.E13] || [[Item:Q4098|<math>\paraU@{-n-\tfrac{1}{2}}{-z} = (-1)^{n}\paraU@{-n-\tfrac{1}{2}}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{-n-\tfrac{1}{2}}{-z} = (-1)^{n}\paraU@{-n-\tfrac{1}{2}}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(- n -(1)/(2), - z) = (- 1)^(n)* CylinderU(- n -(1)/(2), z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(- n -Divide[1,2]), - z] == (- 1)^(n)* ParabolicCylinderD[- 1/2 -(- n -Divide[1,2]), z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
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| | [https://dlmf.nist.gov/12.2.E14 12.2.E14] || [[Item:Q4099|<math>\paraV@{n+\tfrac{1}{2}}{-z} = (-1)^{n}\paraV@{n+\tfrac{1}{2}}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV@{n+\tfrac{1}{2}}{-z} = (-1)^{n}\paraV@{n+\tfrac{1}{2}}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderV(n +(1)/(2), - z) = (- 1)^(n)* CylinderV(n +(1)/(2), z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GAMMA[1/2 + n +Divide[1,2]], Pi]*(Sin[Pi*(n +Divide[1,2])] * ParabolicCylinderD[-(n +Divide[1,2]) - 1/2, - z] + ParabolicCylinderD[-(n +Divide[1,2]) - 1/2, -(- z)]) == (- 1)^(n)* Divide[GAMMA[1/2 + n +Divide[1,2]], Pi]*(Sin[Pi*(n +Divide[1,2])] * ParabolicCylinderD[-(n +Divide[1,2]) - 1/2, z] + ParabolicCylinderD[-(n +Divide[1,2]) - 1/2, -(z)])</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 21]
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| | [https://dlmf.nist.gov/12.2.E15 12.2.E15] || [[Item:Q4100|<math>\paraU@{a}{-z} = -\sin@{\pi a}\paraU@{a}{z}+\frac{\pi}{\EulerGamma@{\frac{1}{2}+a}}\paraV@{a}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{-z} = -\sin@{\pi a}\paraU@{a}{z}+\frac{\pi}{\EulerGamma@{\frac{1}{2}+a}}\paraV@{a}{z}</syntaxhighlight> || <math>\realpart@@{(\frac{1}{2}+a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderU(a, - z) = - sin(Pi*a)*CylinderU(a, z)+(Pi)/(GAMMA((1)/(2)+ a))*CylinderV(a, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), - z] == - Sin[Pi*a]*ParabolicCylinderD[- 1/2 -(a), z]+Divide[Pi,Gamma[Divide[1,2]+ a]]*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)])</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[2.097331412545913, 1.9154557103012664], Times[Complex[-2.097331412545913, -1.9154557103012664], GAMMA[2.0]]]
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| Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.668689589092481, 2.108602350101492], Times[Complex[0.668689589092481, -2.108602350101492], GAMMA[2.0]]]
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| Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.2.E16 12.2.E16] || [[Item:Q4101|<math>\paraV@{a}{-z} = \frac{\cos@{\pi a}}{\EulerGamma@{\frac{1}{2}-a}}\paraU@{a}{z}+\sin@{\pi a}\paraV@{a}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV@{a}{-z} = \frac{\cos@{\pi a}}{\EulerGamma@{\frac{1}{2}-a}}\paraU@{a}{z}+\sin@{\pi a}\paraV@{a}{z}</syntaxhighlight> || <math>\realpart@@{(\frac{1}{2}-a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderV(a, - z) = (cos(Pi*a))/(GAMMA((1)/(2)- a))*CylinderU(a, z)+ sin(Pi*a)*CylinderV(a, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, - z] + ParabolicCylinderD[-(a) - 1/2, -(- z)]) == Divide[Cos[Pi*a],Gamma[Divide[1,2]- a]]*ParabolicCylinderD[- 1/2 -(a), z]+ Sin[Pi*a]*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)])</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.3494376482945125, -0.44804866867585064], Times[Complex[0.1478618109503913, 0.18958829384201614], GAMMA[-1.5]]]
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| Test Values: {Rule[a, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[1.1936070900897449, -0.06991225535058408], Times[Complex[-0.5050655153080368, 0.029582824673347826], GAMMA[-1.5]]]
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| Test Values: {Rule[a, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.2.E17 12.2.E17] || [[Item:Q4102|<math>\sqrt{2\pi}\paraU@{-a}{+ iz} = \EulerGamma@{\tfrac{1}{2}+a}\left(e^{- i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{a}{z}+e^{+ i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{a}{-z}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{2\pi}\paraU@{-a}{+ iz} = \EulerGamma@{\tfrac{1}{2}+a}\left(e^{- i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{a}{z}+e^{+ i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{a}{-z}\right)</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}+a)} > 0</math> || <syntaxhighlight lang=mathematica>sqrt(2*Pi)*CylinderU(- a, + I*z) = GAMMA((1)/(2)+ a)*(exp(- I*Pi*((1)/(2)*a -(1)/(4)))*CylinderU(a, z)+ exp(+ I*Pi*((1)/(2)*a -(1)/(4)))*CylinderU(a, - z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[2*Pi]*ParabolicCylinderD[- 1/2 -(- a), + I*z] == Gamma[Divide[1,2]+ a]*(Exp[- I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[- 1/2 -(a), z]+ Exp[+ I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[- 1/2 -(a), - z])</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
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| | [https://dlmf.nist.gov/12.2.E17 12.2.E17] || [[Item:Q4102|<math>\sqrt{2\pi}\paraU@{-a}{- iz} = \EulerGamma@{\tfrac{1}{2}+a}\left(e^{+ i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{a}{z}+e^{- i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{a}{-z}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{2\pi}\paraU@{-a}{- iz} = \EulerGamma@{\tfrac{1}{2}+a}\left(e^{+ i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{a}{z}+e^{- i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{a}{-z}\right)</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}+a)} > 0</math> || <syntaxhighlight lang=mathematica>sqrt(2*Pi)*CylinderU(- a, - I*z) = GAMMA((1)/(2)+ a)*(exp(+ I*Pi*((1)/(2)*a -(1)/(4)))*CylinderU(a, z)+ exp(- I*Pi*((1)/(2)*a -(1)/(4)))*CylinderU(a, - z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[2*Pi]*ParabolicCylinderD[- 1/2 -(- a), - I*z] == Gamma[Divide[1,2]+ a]*(Exp[+ I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[- 1/2 -(a), z]+ Exp[- I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[- 1/2 -(a), - z])</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
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| | [https://dlmf.nist.gov/12.2.E18 12.2.E18] || [[Item:Q4103|<math>\sqrt{2\pi}\paraU@{a}{z} = \EulerGamma@{\tfrac{1}{2}-a}\left(e^{- i\pi(\frac{1}{2}a+\frac{1}{4})}\paraU@{-a}{+ iz}+e^{+ i\pi(\frac{1}{2}a+\frac{1}{4})}\paraU@{-a}{- iz}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{2\pi}\paraU@{a}{z} = \EulerGamma@{\tfrac{1}{2}-a}\left(e^{- i\pi(\frac{1}{2}a+\frac{1}{4})}\paraU@{-a}{+ iz}+e^{+ i\pi(\frac{1}{2}a+\frac{1}{4})}\paraU@{-a}{- iz}\right)</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}-a)} > 0</math> || <syntaxhighlight lang=mathematica>sqrt(2*Pi)*CylinderU(a, z) = GAMMA((1)/(2)- a)*(exp(- I*Pi*((1)/(2)*a +(1)/(4)))*CylinderU(- a, + I*z)+ exp(+ I*Pi*((1)/(2)*a +(1)/(4)))*CylinderU(- a, - I*z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[2*Pi]*ParabolicCylinderD[- 1/2 -(a), z] == Gamma[Divide[1,2]- a]*(Exp[- I*Pi*(Divide[1,2]*a +Divide[1,4])]*ParabolicCylinderD[- 1/2 -(- a), + I*z]+ Exp[+ I*Pi*(Divide[1,2]*a +Divide[1,4])]*ParabolicCylinderD[- 1/2 -(- a), - I*z])</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
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| | [https://dlmf.nist.gov/12.2.E18 12.2.E18] || [[Item:Q4103|<math>\sqrt{2\pi}\paraU@{a}{z} = \EulerGamma@{\tfrac{1}{2}-a}\left(e^{+ i\pi(\frac{1}{2}a+\frac{1}{4})}\paraU@{-a}{- iz}+e^{- i\pi(\frac{1}{2}a+\frac{1}{4})}\paraU@{-a}{+ iz}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{2\pi}\paraU@{a}{z} = \EulerGamma@{\tfrac{1}{2}-a}\left(e^{+ i\pi(\frac{1}{2}a+\frac{1}{4})}\paraU@{-a}{- iz}+e^{- i\pi(\frac{1}{2}a+\frac{1}{4})}\paraU@{-a}{+ iz}\right)</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}-a)} > 0</math> || <syntaxhighlight lang=mathematica>sqrt(2*Pi)*CylinderU(a, z) = GAMMA((1)/(2)- a)*(exp(+ I*Pi*((1)/(2)*a +(1)/(4)))*CylinderU(- a, - I*z)+ exp(- I*Pi*((1)/(2)*a +(1)/(4)))*CylinderU(- a, + I*z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[2*Pi]*ParabolicCylinderD[- 1/2 -(a), z] == Gamma[Divide[1,2]- a]*(Exp[+ I*Pi*(Divide[1,2]*a +Divide[1,4])]*ParabolicCylinderD[- 1/2 -(- a), - I*z]+ Exp[- I*Pi*(Divide[1,2]*a +Divide[1,4])]*ParabolicCylinderD[- 1/2 -(- a), + I*z])</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
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| | [https://dlmf.nist.gov/12.2.E19 12.2.E19] || [[Item:Q4104|<math>\paraU@{a}{z} = + ie^{+ i\pi a}\paraU@{a}{-z}+\frac{\sqrt{2\pi}}{\EulerGamma@{\tfrac{1}{2}+a}}e^{+ i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{-a}{+ iz}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = + ie^{+ i\pi a}\paraU@{a}{-z}+\frac{\sqrt{2\pi}}{\EulerGamma@{\tfrac{1}{2}+a}}e^{+ i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{-a}{+ iz}</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}+a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = + I*exp(+ I*Pi*a)*CylinderU(a, - z)+(sqrt(2*Pi))/(GAMMA((1)/(2)+ a))*exp(+ I*Pi*((1)/(2)*a -(1)/(4)))*CylinderU(- a, + I*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == + I*Exp[+ I*Pi*a]*ParabolicCylinderD[- 1/2 -(a), - z]+Divide[Sqrt[2*Pi],Gamma[Divide[1,2]+ a]]*Exp[+ I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[- 1/2 -(- a), + I*z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
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| | [https://dlmf.nist.gov/12.2.E19 12.2.E19] || [[Item:Q4104|<math>\paraU@{a}{z} = - ie^{- i\pi a}\paraU@{a}{-z}+\frac{\sqrt{2\pi}}{\EulerGamma@{\tfrac{1}{2}+a}}e^{- i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{-a}{- iz}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = - ie^{- i\pi a}\paraU@{a}{-z}+\frac{\sqrt{2\pi}}{\EulerGamma@{\tfrac{1}{2}+a}}e^{- i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{-a}{- iz}</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}+a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = - I*exp(- I*Pi*a)*CylinderU(a, - z)+(sqrt(2*Pi))/(GAMMA((1)/(2)+ a))*exp(- I*Pi*((1)/(2)*a -(1)/(4)))*CylinderU(- a, - I*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == - I*Exp[- I*Pi*a]*ParabolicCylinderD[- 1/2 -(a), - z]+Divide[Sqrt[2*Pi],Gamma[Divide[1,2]+ a]]*Exp[- I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[- 1/2 -(- a), - I*z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
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| | [https://dlmf.nist.gov/12.2.E20 12.2.E20] || [[Item:Q4105|<math>\paraV@{a}{z} = \frac{- i}{\EulerGamma@{\frac{1}{2}-a}}\paraU@{a}{z}+\sqrt{\frac{2}{\pi}}e^{- i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{-a}{+ iz}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV@{a}{z} = \frac{- i}{\EulerGamma@{\frac{1}{2}-a}}\paraU@{a}{z}+\sqrt{\frac{2}{\pi}}e^{- i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{-a}{+ iz}</syntaxhighlight> || <math>\realpart@@{(\frac{1}{2}-a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderV(a, z) = (- I)/(GAMMA((1)/(2)- a))*CylinderU(a, z)+sqrt((2)/(Pi))*exp(- I*Pi*((1)/(2)*a -(1)/(4)))*CylinderU(- a, + I*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]) == Divide[- I,Gamma[Divide[1,2]- a]]*ParabolicCylinderD[- 1/2 -(a), z]+Sqrt[Divide[2,Pi]]*Exp[- I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[- 1/2 -(- a), + I*z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.4621744673825597, -0.43960813814518984], Times[Complex[3.533949646070574*^-17, 0.0], GAMMA[-1.0]]]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[1.0415095884926804, 0.5968092652227893], Times[Complex[1.0601848938211722*^-16, 3.533949646070574*^-17], GAMMA[-1.0]]]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.2.E20 12.2.E20] || [[Item:Q4105|<math>\paraV@{a}{z} = \frac{+ i}{\EulerGamma@{\frac{1}{2}-a}}\paraU@{a}{z}+\sqrt{\frac{2}{\pi}}e^{+ i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{-a}{- iz}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV@{a}{z} = \frac{+ i}{\EulerGamma@{\frac{1}{2}-a}}\paraU@{a}{z}+\sqrt{\frac{2}{\pi}}e^{+ i\pi(\frac{1}{2}a-\frac{1}{4})}\paraU@{-a}{- iz}</syntaxhighlight> || <math>\realpart@@{(\frac{1}{2}-a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderV(a, z) = (+ I)/(GAMMA((1)/(2)- a))*CylinderU(a, z)+sqrt((2)/(Pi))*exp(+ I*Pi*((1)/(2)*a -(1)/(4)))*CylinderU(- a, - I*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]) == Divide[+ I,Gamma[Divide[1,2]- a]]*ParabolicCylinderD[- 1/2 -(a), z]+Sqrt[Divide[2,Pi]]*Exp[+ I*Pi*(Divide[1,2]*a -Divide[1,4])]*ParabolicCylinderD[- 1/2 -(- a), - I*z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 21] || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.4621744673825599, -0.4396081381451897], Times[Complex[3.533949646070574*^-17, 0.0], GAMMA[-1.0]]]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[1.0415095884926797, 0.5968092652227891], Times[Complex[1.0601848938211722*^-16, 3.533949646070574*^-17], GAMMA[-1.0]]]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.4.E1 12.4.E1] || [[Item:Q4109|<math>\paraU@{a}{z} = \paraU@{a}{0}u_{1}(a,z)+\paraU'@{a}{0}u_{2}(a,z)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = \paraU@{a}{0}u_{1}(a,z)+\paraU'@{a}{0}u_{2}(a,z)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = CylinderU(a, 0)*(exp(-(1)/(4)*(z)^(2))*(1 +(a +(1)/(2))*((z)^(2))/(factorial(2))+(a +(1)/(2))*(a +(5)/(2))*((z)^(4))/(factorial(4))+ ..))+ subs( temp=0, diff( CylinderU(a, temp), temp$(1) ) )*(exp(-(1)/(4)*(z)^(2))*(z +(a +(3)/(2))*((z)^(3))/(factorial(3))+(a +(3)/(2))*(a +(7)/(2))*((z)^(5))/(factorial(5))+ ..))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == ParabolicCylinderD[- 1/2 -(a), 0]*(Exp[-Divide[1,4]*(z)^(2)]*(1 +(a +Divide[1,2])*Divide[(z)^(2),(2)!]+(a +Divide[1,2])*(a +Divide[5,2])*Divide[(z)^(4),(4)!]+ \[Ellipsis]))+ (D[ParabolicCylinderD[- 1/2 -(a), temp], {temp, 1}]/.temp-> 0)*(Exp[-Divide[1,4]*(z)^(2)]*(z +(a +Divide[3,2])*Divide[(z)^(3),(3)!]+(a +Divide[3,2])*(a +Divide[7,2])*Divide[(z)^(5),(5)!]+ \[Ellipsis]))</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.8412106300093095, 0.2667685495532514], Times[Complex[-0.8618940502981999, 0.18957697416081104], Plus[Complex[0.8660254037844387, 0.49999999999999994], …]]]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.7641562690755331, 0.8367141764786244], Times[Complex[-1.1066938670748312, -0.24342165324123666], Plus[Complex[-0.4999999999999998, 0.8660254037844387], …]]]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.4.E2 12.4.E2] || [[Item:Q4110|<math>\paraV@{a}{z} = \paraV@{a}{0}u_{1}(a,z)+\paraV'@{a}{0}u_{2}(a,z)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV@{a}{z} = \paraV@{a}{0}u_{1}(a,z)+\paraV'@{a}{0}u_{2}(a,z)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderV(a, z) = CylinderV(a, 0)*(exp(-(1)/(4)*(z)^(2))*(1 +(a +(1)/(2))*((z)^(2))/(factorial(2))+(a +(1)/(2))*(a +(5)/(2))*((z)^(4))/(factorial(4))+ ..))+ subs( temp=0, diff( CylinderV(a, temp), temp$(1) ) )*(exp(-(1)/(4)*(z)^(2))*(z +(a +(3)/(2))*((z)^(3))/(factorial(3))+(a +(3)/(2))*(a +(7)/(2))*((z)^(5))/(factorial(5))+ ..))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]) == Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, 0] + ParabolicCylinderD[-(a) - 1/2, -(0)])*(Exp[-Divide[1,4]*(z)^(2)]*(1 +(a +Divide[1,2])*Divide[(z)^(2),(2)!]+(a +Divide[1,2])*(a +Divide[5,2])*Divide[(z)^(4),(4)!]+ \[Ellipsis]))+ (D[Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, temp] + ParabolicCylinderD[-(a) - 1/2, -(temp)]), {temp, 1}]/.temp-> 0)*(Exp[-Divide[1,4]*(z)^(2)]*(z +(a +Divide[3,2])*Divide[(z)^(3),(3)!]+(a +Divide[3,2])*(a +Divide[7,2])*Divide[(z)^(5),(5)!]+ \[Ellipsis]))</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [40 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.0, Times[Complex[3.533949646070574*^-17, 0.0], GAMMA[-1.0]]]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[0.0, Times[Complex[1.0601848938211722*^-16, 3.533949646070574*^-17], GAMMA[-1.0]]]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.5.E1 12.5.E1] || [[Item:Q4115|<math>\paraU@{a}{z} = \frac{e^{-\frac{1}{4}z^{2}}}{\EulerGamma@{\frac{1}{2}+a}}\int_{0}^{\infty}t^{a-\frac{1}{2}}e^{-\frac{1}{2}t^{2}-zt}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = \frac{e^{-\frac{1}{4}z^{2}}}{\EulerGamma@{\frac{1}{2}+a}}\int_{0}^{\infty}t^{a-\frac{1}{2}}e^{-\frac{1}{2}t^{2}-zt}\diff{t}</syntaxhighlight> || <math>\realpart@@{a} > -\tfrac{1}{2}, \realpart@@{(\frac{1}{2}+a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = (exp(-(1)/(4)*(z)^(2)))/(GAMMA((1)/(2)+ a))*int((t)^(a -(1)/(2))* exp(-(1)/(2)*(t)^(2)- z*t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == Divide[Exp[-Divide[1,4]*(z)^(2)],Gamma[Divide[1,2]+ a]]*Integrate[(t)^(a -Divide[1,2])* Exp[-Divide[1,2]*(t)^(2)- z*t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 21]
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| | [https://dlmf.nist.gov/12.5.E2 12.5.E2] || [[Item:Q4116|<math>\paraU@{a}{z} = \frac{ze^{-\frac{1}{4}z^{2}}}{\EulerGamma@{\frac{1}{4}+\frac{1}{2}a}}\*\int_{0}^{\infty}t^{\frac{1}{2}a-\frac{3}{4}}e^{-t}\left(z^{2}+2t\right)^{-\frac{1}{2}a-\frac{3}{4}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = \frac{ze^{-\frac{1}{4}z^{2}}}{\EulerGamma@{\frac{1}{4}+\frac{1}{2}a}}\*\int_{0}^{\infty}t^{\frac{1}{2}a-\frac{3}{4}}e^{-t}\left(z^{2}+2t\right)^{-\frac{1}{2}a-\frac{3}{4}}\diff{t}</syntaxhighlight> || <math>|\phase@@{z}| < \tfrac{1}{2}\pi, \realpart@@{a} > -\tfrac{1}{2}, \realpart@@{(\frac{1}{4}+\frac{1}{2}a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = (z*exp(-(1)/(4)*(z)^(2)))/(GAMMA((1)/(4)+(1)/(2)*a))* int((t)^((1)/(2)*a -(3)/(4))* exp(- t)*((z)^(2)+ 2*t)^(-(1)/(2)*a -(3)/(4)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == Divide[z*Exp[-Divide[1,4]*(z)^(2)],Gamma[Divide[1,4]+Divide[1,2]*a]]* Integrate[(t)^(Divide[1,2]*a -Divide[3,4])* Exp[- t]*((z)^(2)+ 2*t)^(-Divide[1,2]*a -Divide[3,4]), {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 15]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
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| Test Values: {a = 2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
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| Test Values: {a = 2, z = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br></div></div> || Successful [Tested: 15]
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| | [https://dlmf.nist.gov/12.5.E3 12.5.E3] || [[Item:Q4117|<math>\paraU@{a}{z} = \frac{e^{-\frac{1}{4}z^{2}}}{\EulerGamma@{\frac{3}{4}+\frac{1}{2}a}}\*\int_{0}^{\infty}t^{\frac{1}{2}a-\frac{1}{4}}e^{-t}\left(z^{2}+2t\right)^{-\frac{1}{2}a-\frac{1}{4}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = \frac{e^{-\frac{1}{4}z^{2}}}{\EulerGamma@{\frac{3}{4}+\frac{1}{2}a}}\*\int_{0}^{\infty}t^{\frac{1}{2}a-\frac{1}{4}}e^{-t}\left(z^{2}+2t\right)^{-\frac{1}{2}a-\frac{1}{4}}\diff{t}</syntaxhighlight> || <math>|\phase@@{z}| < \tfrac{1}{2}\pi, \realpart@@{a} > -\tfrac{3}{2}, \realpart@@{(\frac{3}{4}+\frac{1}{2}a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = (exp(-(1)/(4)*(z)^(2)))/(GAMMA((3)/(4)+(1)/(2)*a))* int((t)^((1)/(2)*a -(1)/(4))* exp(- t)*((z)^(2)+ 2*t)^(-(1)/(2)*a -(1)/(4)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == Divide[Exp[-Divide[1,4]*(z)^(2)],Gamma[Divide[3,4]+Divide[1,2]*a]]* Integrate[(t)^(Divide[1,2]*a -Divide[1,4])* Exp[- t]*((z)^(2)+ 2*t)^(-Divide[1,2]*a -Divide[1,4]), {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 20]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
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| Test Values: {a = 3/2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
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| Test Values: {a = 3/2, z = 1/2-1/2*I*3^(1/2)}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [5 / 20]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Indeterminate
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| Test Values: {Rule[a, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Indeterminate
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| Test Values: {Rule[a, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.5.E4 12.5.E4] || [[Item:Q4118|<math>\paraU@{a}{z} = \sqrt{\frac{2}{\pi}}e^{\frac{1}{4}z^{2}}\*\int_{0}^{\infty}t^{-a-\frac{1}{2}}e^{-\frac{1}{2}t^{2}}\cos@{zt+\left(\tfrac{1}{2}a+\tfrac{1}{4}\right)\pi}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = \sqrt{\frac{2}{\pi}}e^{\frac{1}{4}z^{2}}\*\int_{0}^{\infty}t^{-a-\frac{1}{2}}e^{-\frac{1}{2}t^{2}}\cos@{zt+\left(\tfrac{1}{2}a+\tfrac{1}{4}\right)\pi}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = sqrt((2)/(Pi))*exp((1)/(4)*(z)^(2))* int((t)^(- a -(1)/(2))* exp(-(1)/(2)*(t)^(2))*cos(z*t +((1)/(2)*a +(1)/(4))*Pi), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == Sqrt[Divide[2,Pi]]*Exp[Divide[1,4]*(z)^(2)]* Integrate[(t)^(- a -Divide[1,2])* Exp[-Divide[1,2]*(t)^(2)]*Cos[z*t +(Divide[1,2]*a +Divide[1,4])*Pi], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 7]
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| | [https://dlmf.nist.gov/12.5.E5 12.5.E5] || [[Item:Q4119|<math>\paraU@{a}{z} = \frac{\EulerGamma@{\frac{1}{2}-a}}{2\pi i}e^{-\frac{1}{4}z^{2}}\int_{-\infty}^{(0+)}e^{zt-\frac{1}{2}t^{2}}t^{a-\frac{1}{2}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = \frac{\EulerGamma@{\frac{1}{2}-a}}{2\pi i}e^{-\frac{1}{4}z^{2}}\int_{-\infty}^{(0+)}e^{zt-\frac{1}{2}t^{2}}t^{a-\frac{1}{2}}\diff{t}</syntaxhighlight> || <math>a \neq \frac{1}{2}, -\pi < \phase@@{t}, \phase@@{t} < \pi, \realpart@@{(\frac{1}{2}-a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = (GAMMA((1)/(2)- a))/(2*Pi*I)*exp(-(1)/(4)*(z)^(2))*int(exp(z*t -(1)/(2)*(t)^(2))*(t)^(a -(1)/(2)), t = - infinity..(0 +))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == Divide[Gamma[Divide[1,2]- a],2*Pi*I]*Exp[-Divide[1,4]*(z)^(2)]*Integrate[Exp[z*t -Divide[1,2]*(t)^(2)]*(t)^(a -Divide[1,2]), {t, - Infinity, (0 +)}, GenerateConditions->None]</syntaxhighlight> || Error || Failure || - || Error
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| | [https://dlmf.nist.gov/12.5.E6 12.5.E6] || [[Item:Q4120|<math>\paraU@{a}{z} = \frac{e^{\frac{1}{4}z^{2}}}{i\sqrt{2\pi}}\int_{c-i\infty}^{c+i\infty}e^{-zt+\frac{1}{2}t^{2}}t^{-a-\frac{1}{2}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = \frac{e^{\frac{1}{4}z^{2}}}{i\sqrt{2\pi}}\int_{c-i\infty}^{c+i\infty}e^{-zt+\frac{1}{2}t^{2}}t^{-a-\frac{1}{2}}\diff{t}</syntaxhighlight> || <math>-\tfrac{1}{2}\pi < \phase@@{t}, \phase@@{t} < \tfrac{1}{2}\pi, c > 0</math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = (exp((1)/(4)*(z)^(2)))/(I*sqrt(2*Pi))*int(exp(- z*t +(1)/(2)*(t)^(2))*(t)^(- a -(1)/(2)), t = c - I*infinity..c + I*infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == Divide[Exp[Divide[1,4]*(z)^(2)],I*Sqrt[2*Pi]]*Integrate[Exp[- z*t +Divide[1,2]*(t)^(2)]*(t)^(- a -Divide[1,2]), {t, c - I*Infinity, c + I*Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [126 / 126]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .8412106295+.2667685493*I
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| Test Values: {a = -3/2, c = 3/2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.7641562685+.8367141760*I
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| Test Values: {a = -3/2, c = 3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
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| | [https://dlmf.nist.gov/12.5.E8 12.5.E8] || [[Item:Q4122|<math>\paraU@{a}{z} = \frac{e^{-\frac{1}{4}z^{2}}z^{-a-\frac{1}{2}}}{2\pi i\EulerGamma@{\frac{1}{2}+a}}\*\int_{-i\infty}^{i\infty}\EulerGamma@{t}\EulerGamma@{\tfrac{1}{2}+a-2t}2^{t}z^{2t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = \frac{e^{-\frac{1}{4}z^{2}}z^{-a-\frac{1}{2}}}{2\pi i\EulerGamma@{\frac{1}{2}+a}}\*\int_{-i\infty}^{i\infty}\EulerGamma@{t}\EulerGamma@{\tfrac{1}{2}+a-2t}2^{t}z^{2t}\diff{t}</syntaxhighlight> || <math>a \neq -\frac{1}{2}, |\phase@@{z}| < \tfrac{3}{4}\pi, \realpart@@{t} > 0, \realpart@@{(\tfrac{1}{2}+a-2t)} > 0, \realpart@@{(\frac{1}{2}+a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = (exp(-(1)/(4)*(z)^(2))*(z)^(- a -(1)/(2)))/(2*Pi*I*GAMMA((1)/(2)+ a))* int(GAMMA(t)*GAMMA((1)/(2)+ a - 2*t)*(2)^(t)* (z)^(2*t), t = - I*infinity..I*infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == Divide[Exp[-Divide[1,4]*(z)^(2)]*(z)^(- a -Divide[1,2]),2*Pi*I*Gamma[Divide[1,2]+ a]]* Integrate[Gamma[t]*Gamma[Divide[1,2]+ a - 2*t]*(2)^(t)* (z)^(2*t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/12.5.E9 12.5.E9] || [[Item:Q4123|<math>\paraV@{a}{z} = \sqrt{\frac{2}{\pi}}\frac{e^{\frac{1}{4}z^{2}}z^{a-\frac{1}{2}}}{2\pi i\EulerGamma@{\frac{1}{2}-a}}\*\int_{-i\infty}^{i\infty}\EulerGamma@{t}\EulerGamma@{\tfrac{1}{2}-a-2t}2^{t}z^{2t}\cos@{\pi t}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV@{a}{z} = \sqrt{\frac{2}{\pi}}\frac{e^{\frac{1}{4}z^{2}}z^{a-\frac{1}{2}}}{2\pi i\EulerGamma@{\frac{1}{2}-a}}\*\int_{-i\infty}^{i\infty}\EulerGamma@{t}\EulerGamma@{\tfrac{1}{2}-a-2t}2^{t}z^{2t}\cos@{\pi t}\diff{t}</syntaxhighlight> || <math>a \neq \frac{1}{2}, |\phase@@{z}| < \tfrac{1}{4}\pi, \realpart@@{t} > 0, \realpart@@{(\tfrac{1}{2}-a-2t)} > 0, \realpart@@{(\frac{1}{2}-a)} > 0</math> || <syntaxhighlight lang=mathematica>CylinderV(a, z) = sqrt((2)/(Pi))*(exp((1)/(4)*(z)^(2))*(z)^(a -(1)/(2)))/(2*Pi*I*GAMMA((1)/(2)- a))* int(GAMMA(t)*GAMMA((1)/(2)- a - 2*t)*(2)^(t)* (z)^(2*t)* cos(Pi*t), t = - I*infinity..I*infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]) == Sqrt[Divide[2,Pi]]*Divide[Exp[Divide[1,4]*(z)^(2)]*(z)^(a -Divide[1,2]),2*Pi*I*Gamma[Divide[1,2]- a]]* Integrate[Gamma[t]*Gamma[Divide[1,2]- a - 2*t]*(2)^(t)* (z)^(2*t)* Cos[Pi*t], {t, - I*Infinity, I*Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/12.7.E1 12.7.E1] || [[Item:Q4124|<math>\paraU@{-\tfrac{1}{2}}{z} = \WhittakerparaD{0}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{-\tfrac{1}{2}}{z} = \WhittakerparaD{0}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(-(1)/(2), z) = CylinderD(0, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(-Divide[1,2]), z] == ParabolicCylinderD[0, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| | [https://dlmf.nist.gov/12.7.E1 12.7.E1] || [[Item:Q4124|<math>\WhittakerparaD{0}@{z} = e^{-\frac{1}{4}z^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerparaD{0}@{z} = e^{-\frac{1}{4}z^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderD(0, z) = exp(-(1)/(4)*(z)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[0, z] == Exp[-Divide[1,4]*(z)^(2)]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| | [https://dlmf.nist.gov/12.7.E2 12.7.E2] || [[Item:Q4125|<math>\paraU@{-n-\tfrac{1}{2}}{z} = \WhittakerparaD{n}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{-n-\tfrac{1}{2}}{z} = \WhittakerparaD{n}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(- n -(1)/(2), z) = CylinderD(n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(- n -Divide[1,2]), z] == ParabolicCylinderD[n, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| | [https://dlmf.nist.gov/12.7.E4 12.7.E4] || [[Item:Q4127|<math>\paraV@{-\tfrac{1}{2}}{z} = (\ifrac{2}{\sqrt{\pi}}\,)e^{\frac{1}{4}z^{2}}\DawsonsintF@{z/\sqrt{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV@{-\tfrac{1}{2}}{z} = (\ifrac{2}{\sqrt{\pi}}\,)e^{\frac{1}{4}z^{2}}\DawsonsintF@{z/\sqrt{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderV(-(1)/(2), z) = ((2)/(sqrt(Pi)))*exp((1)/(4)*(z)^(2))*dawson(z/(sqrt(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GAMMA[1/2 + -Divide[1,2]], Pi]*(Sin[Pi*(-Divide[1,2])] * ParabolicCylinderD[-(-Divide[1,2]) - 1/2, z] + ParabolicCylinderD[-(-Divide[1,2]) - 1/2, -(z)]) == (Divide[2,Sqrt[Pi]])*Exp[Divide[1,4]*(z)^(2)]*DawsonF[z/(Sqrt[2])]</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.6813729414422256, -0.33849358809725466]
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| Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.4709386394349885, -0.6804499612300876]
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| Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.7.E5 12.7.E5] || [[Item:Q4128|<math>\paraU@{\tfrac{1}{2}}{z} = \WhittakerparaD{-1}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{\tfrac{1}{2}}{z} = \WhittakerparaD{-1}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU((1)/(2), z) = CylinderD(- 1, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(Divide[1,2]), z] == ParabolicCylinderD[- 1, z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| | [https://dlmf.nist.gov/12.7.E5 12.7.E5] || [[Item:Q4128|<math>\WhittakerparaD{-1}@{z} = \sqrt{\tfrac{1}{2}\pi}\,e^{\frac{1}{4}z^{2}}\erfc@{z/\sqrt{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerparaD{-1}@{z} = \sqrt{\tfrac{1}{2}\pi}\,e^{\frac{1}{4}z^{2}}\erfc@{z/\sqrt{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderD(- 1, z) = sqrt((1)/(2)*Pi)*exp((1)/(4)*(z)^(2))*erfc(z/(sqrt(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1, z] == Sqrt[Divide[1,2]*Pi]*Exp[Divide[1,4]*(z)^(2)]*Erfc[z/(Sqrt[2])]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| | [https://dlmf.nist.gov/12.7.E6 12.7.E6] || [[Item:Q4129|<math>\paraU@{n+\tfrac{1}{2}}{z} = \WhittakerparaD{-n-1}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{n+\tfrac{1}{2}}{z} = \WhittakerparaD{-n-1}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(n +(1)/(2), z) = CylinderD(- n - 1, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(n +Divide[1,2]), z] == ParabolicCylinderD[- n - 1, z]</syntaxhighlight> || Successful || Successful || Manual Skip! || Successful [Tested: 7]
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| | [https://dlmf.nist.gov/12.7.E6 12.7.E6] || [[Item:Q4129|<math>\WhittakerparaD{-n-1}@{z} = \sqrt{\frac{\pi}{2}}\frac{(-1)^{n}}{n!}e^{-\frac{1}{4}z^{2}}\deriv[n]{\left(e^{\frac{1}{2}z^{2}}\erfc@{z/\sqrt{2}}\right)}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerparaD{-n-1}@{z} = \sqrt{\frac{\pi}{2}}\frac{(-1)^{n}}{n!}e^{-\frac{1}{4}z^{2}}\deriv[n]{\left(e^{\frac{1}{2}z^{2}}\erfc@{z/\sqrt{2}}\right)}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderD(- n - 1, z) = sqrt((Pi)/(2))*((- 1)^(n))/(factorial(n))*exp(-(1)/(4)*(z)^(2))*diff(exp((1)/(2)*(z)^(2))*erfc(z/(sqrt(2))), [z$(n)])</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- n - 1, z] == Sqrt[Divide[Pi,2]]*Divide[(- 1)^(n),(n)!]*Exp[-Divide[1,4]*(z)^(2)]*D[Exp[Divide[1,2]*(z)^(2)]*Erfc[z/(Sqrt[2])], {z, n}]</syntaxhighlight> || Failure || Failure || Manual Skip! || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.017848622575954935, Times[0.7141168694348256, DifferenceRoot[Function[{, }
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| Test Values: {Equal[Plus[Times[-1, []], Times[-1, 1.5, [Plus[1, ]]], Times[Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 2], Power[1.5, 2]]], Erfc[Times[Power[2, Rational[-1, 2]], 1.5]]]], Equal[[1], Plus[Times[-1, Power[Times[2, Power[Pi, -1]], Rational[1, 2]]], Times[Power[E, Times[Rational[1, 2], Power[1.5, 2]]], 1.5, Erfc[Times[Power[2, Rational[-1, 2]], 1.5]]]]]}]][3.0]]], {Rule[n, 3], Rule[z, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[0.1293114227985036, Times[1.1773796724029832, DifferenceRoot[Function[{, }
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| Test Values: {Equal[Plus[Times[-1, []], Times[-1, 0.5, [Plus[1, ]]], Times[Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 2], Power[0.5, 2]]], Erfc[Times[Power[2, Rational[-1, 2]], 0.5]]]], Equal[[1], Plus[Times[-1, Power[Times[2, Power[Pi, -1]], Rational[1, 2]]], Times[Power[E, Times[Rational[1, 2], Power[0.5, 2]]], 0.5, Erfc[Times[Power[2, Rational[-1, 2]], 0.5]]]]]}]][3.0]]], {Rule[n, 3], Rule[z, 0.5]}</syntaxhighlight><br></div></div>
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| | [https://dlmf.nist.gov/12.7.E8 12.7.E8] || [[Item:Q4131|<math>\paraU@{-2}{z} = \frac{z^{5/2}}{4\sqrt{2\pi}}\left(2\modBesselK{\frac{1}{4}}@{\tfrac{1}{4}z^{2}}+3\modBesselK{\frac{3}{4}}@{\tfrac{1}{4}z^{2}}-\modBesselK{\frac{5}{4}}@{\tfrac{1}{4}z^{2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{-2}{z} = \frac{z^{5/2}}{4\sqrt{2\pi}}\left(2\modBesselK{\frac{1}{4}}@{\tfrac{1}{4}z^{2}}+3\modBesselK{\frac{3}{4}}@{\tfrac{1}{4}z^{2}}-\modBesselK{\frac{5}{4}}@{\tfrac{1}{4}z^{2}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(- 2, z) = ((z)^(5/2))/(4*sqrt(2*Pi))*(2*BesselK((1)/(4), (1)/(4)*(z)^(2))+ 3*BesselK((3)/(4), (1)/(4)*(z)^(2))- BesselK((5)/(4), (1)/(4)*(z)^(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(- 2), z] == Divide[(z)^(5/2),4*Sqrt[2*Pi]]*(2*BesselK[Divide[1,4], Divide[1,4]*(z)^(2)]+ 3*BesselK[Divide[3,4], Divide[1,4]*(z)^(2)]- BesselK[Divide[5,4], Divide[1,4]*(z)^(2)])</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -2.928712959+.1903824416*I
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| Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.578570932+.7263102924*I
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| Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-2.928712959362369, 0.19038244130086163]
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| Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.5785709321816723, 0.7263102922437361]
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| Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div>
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| |-
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| | [https://dlmf.nist.gov/12.7.E9 12.7.E9] || [[Item:Q4132|<math>\paraU@{-1}{z} = \frac{z^{3/2}}{2\sqrt{2\pi}}\left(\modBesselK{\frac{1}{4}}@{\tfrac{1}{4}z^{2}}+\modBesselK{\frac{3}{4}}@{\tfrac{1}{4}z^{2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{-1}{z} = \frac{z^{3/2}}{2\sqrt{2\pi}}\left(\modBesselK{\frac{1}{4}}@{\tfrac{1}{4}z^{2}}+\modBesselK{\frac{3}{4}}@{\tfrac{1}{4}z^{2}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(- 1, z) = ((z)^(3/2))/(2*sqrt(2*Pi))*(BesselK((1)/(4), (1)/(4)*(z)^(2))+ BesselK((3)/(4), (1)/(4)*(z)^(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(- 1), z] == Divide[(z)^(3/2),2*Sqrt[2*Pi]]*(BesselK[Divide[1,4], Divide[1,4]*(z)^(2)]+ BesselK[Divide[3,4], Divide[1,4]*(z)^(2)])</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .5254625443+1.913964596*I
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| Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.1061142274-1.367750447*I
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| Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.5254625445137794, 1.9139645960722755]
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| Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.10611422720224939, -1.3677504477251]
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| Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div>
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| | [https://dlmf.nist.gov/12.7.E10 12.7.E10] || [[Item:Q4133|<math>\paraU@{0}{z} = \sqrt{\frac{z}{2\pi}}\modBesselK{\frac{1}{4}}@{\tfrac{1}{4}z^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{0}{z} = \sqrt{\frac{z}{2\pi}}\modBesselK{\frac{1}{4}}@{\tfrac{1}{4}z^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(0, z) = sqrt((z)/(2*Pi))*BesselK((1)/(4), (1)/(4)*(z)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(0), z] == Sqrt[Divide[z,2*Pi]]*BesselK[Divide[1,4], Divide[1,4]*(z)^(2)]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 2.016879450-1.384601654*I
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| Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.973186649+1.022506910*I
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| Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[2.0168794499257325, -1.3846016541017099]
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| Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.9731866495584476, 1.0225069102497304]
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| Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div>
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| | [https://dlmf.nist.gov/12.7.E11 12.7.E11] || [[Item:Q4134|<math>\paraU@{1}{z} = \frac{z^{3/2}}{\sqrt{2\pi}}\left(\modBesselK{\frac{3}{4}}@{\tfrac{1}{4}z^{2}}-\modBesselK{\frac{1}{4}}@{\tfrac{1}{4}z^{2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{1}{z} = \frac{z^{3/2}}{\sqrt{2\pi}}\left(\modBesselK{\frac{3}{4}}@{\tfrac{1}{4}z^{2}}-\modBesselK{\frac{1}{4}}@{\tfrac{1}{4}z^{2}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(1, z) = ((z)^(3/2))/(sqrt(2*Pi))*(BesselK((3)/(4), (1)/(4)*(z)^(2))- BesselK((1)/(4), (1)/(4)*(z)^(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(1), z] == Divide[(z)^(3/2),Sqrt[2*Pi]]*(BesselK[Divide[3,4], Divide[1,4]*(z)^(2)]- BesselK[Divide[1,4], Divide[1,4]*(z)^(2)])</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .6696041257-1.050010143*I
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| Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 2.182924166+1.008719675*I
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| Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.6696041258052213, -1.050010141970097]
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| Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[2.1829241651976083, 1.0087196737510498]
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| Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div>
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| | [https://dlmf.nist.gov/12.7.E14 12.7.E14] || [[Item:Q4137|<math>\paraU@{a}{z} = 2^{-\frac{1}{4}-\frac{1}{2}a}e^{-\frac{1}{4}z^{2}}\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{1}{4}}{\tfrac{1}{2}}{\tfrac{1}{2}z^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{z} = 2^{-\frac{1}{4}-\frac{1}{2}a}e^{-\frac{1}{4}z^{2}}\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{1}{4}}{\tfrac{1}{2}}{\tfrac{1}{2}z^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(a, z) = (2)^(-(1)/(4)-(1)/(2)*a)* exp(-(1)/(4)*(z)^(2))*KummerU((1)/(2)*a +(1)/(4), (1)/(2), (1)/(2)*(z)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), z] == (2)^(-Divide[1,4]-Divide[1,2]*a)* Exp[-Divide[1,4]*(z)^(2)]*HypergeometricU[Divide[1,2]*a +Divide[1,4], Divide[1,2], Divide[1,2]*(z)^(2)]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.528312538+1.673428352*I
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| Test Values: {a = -3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.682421259-.5335370987*I
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| Test Values: {a = -3/2, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.5283125381510665, 1.6734283529572487]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.6824212600186188, -0.5335370991065028]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.7.E14 12.7.E14] || [[Item:Q4137|<math>2^{-\frac{1}{4}-\frac{1}{2}a}e^{-\frac{1}{4}z^{2}}\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{1}{4}}{\tfrac{1}{2}}{\tfrac{1}{2}z^{2}} = 2^{-\frac{3}{4}-\frac{1}{2}a}ze^{-\frac{1}{4}z^{2}}\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{3}{4}}{\tfrac{3}{2}}{\tfrac{1}{2}z^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>2^{-\frac{1}{4}-\frac{1}{2}a}e^{-\frac{1}{4}z^{2}}\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{1}{4}}{\tfrac{1}{2}}{\tfrac{1}{2}z^{2}} = 2^{-\frac{3}{4}-\frac{1}{2}a}ze^{-\frac{1}{4}z^{2}}\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{3}{4}}{\tfrac{3}{2}}{\tfrac{1}{2}z^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(2)^(-(1)/(4)-(1)/(2)*a)* exp(-(1)/(4)*(z)^(2))*KummerU((1)/(2)*a +(1)/(4), (1)/(2), (1)/(2)*(z)^(2)) = (2)^(-(3)/(4)-(1)/(2)*a)* z*exp(-(1)/(4)*(z)^(2))*KummerU((1)/(2)*a +(3)/(4), (3)/(2), (1)/(2)*(z)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(2)^(-Divide[1,4]-Divide[1,2]*a)* Exp[-Divide[1,4]*(z)^(2)]*HypergeometricU[Divide[1,2]*a +Divide[1,4], Divide[1,2], Divide[1,2]*(z)^(2)] == (2)^(-Divide[3,4]-Divide[1,2]*a)* z*Exp[-Divide[1,4]*(z)^(2)]*HypergeometricU[Divide[1,2]*a +Divide[3,4], Divide[3,2], Divide[1,2]*(z)^(2)]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.528312538-1.673428353*I
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| Test Values: {a = -3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.682421260+.5335370988*I
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| Test Values: {a = -3/2, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.5283125381510665, -1.673428352957249]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.6824212600186188, 0.5335370991065027]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.7.E14 12.7.E14] || [[Item:Q4137|<math>2^{-\frac{3}{4}-\frac{1}{2}a}ze^{-\frac{1}{4}z^{2}}\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{3}{4}}{\tfrac{3}{2}}{\tfrac{1}{2}z^{2}} = 2^{-\frac{1}{2}a}z^{-\frac{1}{2}}\WhittakerconfhyperW{-\frac{1}{2}a}{+\frac{1}{4}}@{\tfrac{1}{2}z^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>2^{-\frac{3}{4}-\frac{1}{2}a}ze^{-\frac{1}{4}z^{2}}\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{3}{4}}{\tfrac{3}{2}}{\tfrac{1}{2}z^{2}} = 2^{-\frac{1}{2}a}z^{-\frac{1}{2}}\WhittakerconfhyperW{-\frac{1}{2}a}{+\frac{1}{4}}@{\tfrac{1}{2}z^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(2)^(-(3)/(4)-(1)/(2)*a)* z*exp(-(1)/(4)*(z)^(2))*KummerU((1)/(2)*a +(3)/(4), (3)/(2), (1)/(2)*(z)^(2)) = (2)^(-(1)/(2)*a)* (z)^(-(1)/(2))* WhittakerW(-(1)/(2)*a, +(1)/(4), (1)/(2)*(z)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(2)^(-Divide[3,4]-Divide[1,2]*a)* z*Exp[-Divide[1,4]*(z)^(2)]*HypergeometricU[Divide[1,2]*a +Divide[3,4], Divide[3,2], Divide[1,2]*(z)^(2)] == (2)^(-Divide[1,2]*a)* (z)^(-Divide[1,2])* WhittakerW[-Divide[1,2]*a, +Divide[1,4], Divide[1,2]*(z)^(2)]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .725579081e-1+1.600870446*I
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| Test Values: {a = -3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.5744420805-1.107979180*I
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| Test Values: {a = -3/2, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.0725579074030912, 1.600870445554158]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.574442080456058, -1.1079791795625606]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.7.E14 12.7.E14] || [[Item:Q4137|<math>2^{-\frac{3}{4}-\frac{1}{2}a}ze^{-\frac{1}{4}z^{2}}\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{3}{4}}{\tfrac{3}{2}}{\tfrac{1}{2}z^{2}} = 2^{-\frac{1}{2}a}z^{-\frac{1}{2}}\WhittakerconfhyperW{-\frac{1}{2}a}{-\frac{1}{4}}@{\tfrac{1}{2}z^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>2^{-\frac{3}{4}-\frac{1}{2}a}ze^{-\frac{1}{4}z^{2}}\KummerconfhyperU@{\tfrac{1}{2}a+\tfrac{3}{4}}{\tfrac{3}{2}}{\tfrac{1}{2}z^{2}} = 2^{-\frac{1}{2}a}z^{-\frac{1}{2}}\WhittakerconfhyperW{-\frac{1}{2}a}{-\frac{1}{4}}@{\tfrac{1}{2}z^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(2)^(-(3)/(4)-(1)/(2)*a)* z*exp(-(1)/(4)*(z)^(2))*KummerU((1)/(2)*a +(3)/(4), (3)/(2), (1)/(2)*(z)^(2)) = (2)^(-(1)/(2)*a)* (z)^(-(1)/(2))* WhittakerW(-(1)/(2)*a, -(1)/(4), (1)/(2)*(z)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(2)^(-Divide[3,4]-Divide[1,2]*a)* z*Exp[-Divide[1,4]*(z)^(2)]*HypergeometricU[Divide[1,2]*a +Divide[3,4], Divide[3,2], Divide[1,2]*(z)^(2)] == (2)^(-Divide[1,2]*a)* (z)^(-Divide[1,2])* WhittakerW[-Divide[1,2]*a, -Divide[1,4], Divide[1,2]*(z)^(2)]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .725579081e-1+1.600870446*I
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| Test Values: {a = -3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.5744420804-1.107979180*I
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| Test Values: {a = -3/2, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.0725579074030912, 1.600870445554158]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.5744420804560579, -1.1079791795625609]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.8.E1 12.8.E1] || [[Item:Q4138|<math>z\paraU@{a}{z}-\paraU@{a-1}{z}+(a+\tfrac{1}{2})\paraU@{a+1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z\paraU@{a}{z}-\paraU@{a-1}{z}+(a+\tfrac{1}{2})\paraU@{a+1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>z*CylinderU(a, z)- CylinderU(a - 1, z)+(a +(1)/(2))*CylinderU(a + 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>z*ParabolicCylinderD[- 1/2 -(a), z]- ParabolicCylinderD[- 1/2 -(a - 1), z]+(a +Divide[1,2])*ParabolicCylinderD[- 1/2 -(a + 1), z] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 42]
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| | [https://dlmf.nist.gov/12.8.E2 12.8.E2] || [[Item:Q4139|<math>\paraU'@{a}{z}+\tfrac{1}{2}z\paraU@{a}{z}+(a+\tfrac{1}{2})\paraU@{a+1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU'@{a}{z}+\tfrac{1}{2}z\paraU@{a}{z}+(a+\tfrac{1}{2})\paraU@{a+1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff( CylinderU(a, z), z$(1) )+(1)/(2)*z*CylinderU(a, z)+(a +(1)/(2))*CylinderU(a + 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[ParabolicCylinderD[- 1/2 -(a), z], {z, 1}]+Divide[1,2]*z*ParabolicCylinderD[- 1/2 -(a), z]+(a +Divide[1,2])*ParabolicCylinderD[- 1/2 -(a + 1), z] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 42]
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| | [https://dlmf.nist.gov/12.8.E3 12.8.E3] || [[Item:Q4140|<math>\paraU'@{a}{z}-\tfrac{1}{2}z\paraU@{a}{z}+\paraU@{a-1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU'@{a}{z}-\tfrac{1}{2}z\paraU@{a}{z}+\paraU@{a-1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff( CylinderU(a, z), z$(1) )-(1)/(2)*z*CylinderU(a, z)+ CylinderU(a - 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[ParabolicCylinderD[- 1/2 -(a), z], {z, 1}]-Divide[1,2]*z*ParabolicCylinderD[- 1/2 -(a), z]+ ParabolicCylinderD[- 1/2 -(a - 1), z] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 42]
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| | [https://dlmf.nist.gov/12.8.E4 12.8.E4] || [[Item:Q4141|<math>2\paraU'@{a}{z}+\paraU@{a-1}{z}+(a+\tfrac{1}{2})\paraU@{a+1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>2\paraU'@{a}{z}+\paraU@{a-1}{z}+(a+\tfrac{1}{2})\paraU@{a+1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>2*diff( CylinderU(a, z), z$(1) )+ CylinderU(a - 1, z)+(a +(1)/(2))*CylinderU(a + 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>2*D[ParabolicCylinderD[- 1/2 -(a), z], {z, 1}]+ ParabolicCylinderD[- 1/2 -(a - 1), z]+(a +Divide[1,2])*ParabolicCylinderD[- 1/2 -(a + 1), z] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 42]
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| | [https://dlmf.nist.gov/12.8.E5 12.8.E5] || [[Item:Q4142|<math>z\paraV@{a}{z}-\paraV@{a+1}{z}+(a-\tfrac{1}{2})\paraV@{a-1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>z\paraV@{a}{z}-\paraV@{a+1}{z}+(a-\tfrac{1}{2})\paraV@{a-1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>z*CylinderV(a, z)- CylinderV(a + 1, z)+(a -(1)/(2))*CylinderV(a - 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>z*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)])- Divide[GAMMA[1/2 + a + 1], Pi]*(Sin[Pi*(a + 1)] * ParabolicCylinderD[-(a + 1) - 1/2, z] + ParabolicCylinderD[-(a + 1) - 1/2, -(z)])+(a -Divide[1,2])*Divide[GAMMA[1/2 + a - 1], Pi]*(Sin[Pi*(a - 1)] * ParabolicCylinderD[-(a - 1) - 1/2, z] + ParabolicCylinderD[-(a - 1) - 1/2, -(z)]) == 0</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[7.067899292141149*^-17, 0.0], GAMMA[-2.0]], Times[Complex[3.060490169192143*^-17, 1.7669748230352868*^-17], GAMMA[-1.0]], Times[Complex[0.0, -8.834874115176436*^-18], GAMMA[0.0]]]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[1.4135798584282297*^-16, 0.0], GAMMA[-2.0]], Times[Complex[-8.361414638298002*^-17, 7.414495684541142*^-17], GAMMA[-1.0]], Times[Complex[-7.067899292141149*^-17, -8.834874115176436*^-18], GAMMA[0.0]]]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.8.E6 12.8.E6] || [[Item:Q4143|<math>\paraV'@{a}{z}-\tfrac{1}{2}z\paraV@{a}{z}-(a-\tfrac{1}{2})\paraV@{a-1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV'@{a}{z}-\tfrac{1}{2}z\paraV@{a}{z}-(a-\tfrac{1}{2})\paraV@{a-1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff( CylinderV(a, z), z$(1) )-(1)/(2)*z*CylinderV(a, z)-(a -(1)/(2))*CylinderV(a - 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]), {z, 1}]-Divide[1,2]*z*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)])-(a -Divide[1,2])*Divide[GAMMA[1/2 + a - 1], Pi]*(Sin[Pi*(a - 1)] * ParabolicCylinderD[-(a - 1) - 1/2, z] + ParabolicCylinderD[-(a - 1) - 1/2, -(z)]) == 0</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [39 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.0, 0.0], Times[Complex[-7.067899292141149*^-17, 0.0], GAMMA[-2.0]], Times[Complex[-1.5302450845960716*^-17, -8.834874115176434*^-18], GAMMA[-1.0]]]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[-1.4135798584282297*^-16, 0.0], GAMMA[-2.0]], Times[Complex[-9.955091265133296*^-17, -1.7329819619999673*^-18], GAMMA[-1.0]]]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.8.E7 12.8.E7] || [[Item:Q4144|<math>\paraV'@{a}{z}+\tfrac{1}{2}z\paraV@{a}{z}-\paraV@{a+1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV'@{a}{z}+\tfrac{1}{2}z\paraV@{a}{z}-\paraV@{a+1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff( CylinderV(a, z), z$(1) )+(1)/(2)*z*CylinderV(a, z)- CylinderV(a + 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]), {z, 1}]+Divide[1,2]*z*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)])- Divide[GAMMA[1/2 + a + 1], Pi]*(Sin[Pi*(a + 1)] * ParabolicCylinderD[-(a + 1) - 1/2, z] + ParabolicCylinderD[-(a + 1) - 1/2, -(z)]) == 0</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.0, 0.0], Times[Complex[1.5302450845960716*^-17, 8.834874115176434*^-18], GAMMA[-1.0]], Times[Complex[0.0, -8.834874115176436*^-18], GAMMA[0.0]]]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[-1.83165059034313*^-16, 7.241197488341145*^-17], GAMMA[-1.0]], Times[Complex[-7.067899292141149*^-17, -8.834874115176436*^-18], GAMMA[0.0]]]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.8.E8 12.8.E8] || [[Item:Q4145|<math>2\paraV'@{a}{z}-\paraV@{a+1}{z}-(a-\tfrac{1}{2})\paraV@{a-1}{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>2\paraV'@{a}{z}-\paraV@{a+1}{z}-(a-\tfrac{1}{2})\paraV@{a-1}{z} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>2*diff( CylinderV(a, z), z$(1) )- CylinderV(a + 1, z)-(a -(1)/(2))*CylinderV(a - 1, z) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>2*D[Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]), {z, 1}]- Divide[GAMMA[1/2 + a + 1], Pi]*(Sin[Pi*(a + 1)] * ParabolicCylinderD[-(a + 1) - 1/2, z] + ParabolicCylinderD[-(a + 1) - 1/2, -(z)])-(a -Divide[1,2])*Divide[GAMMA[1/2 + a - 1], Pi]*(Sin[Pi*(a - 1)] * ParabolicCylinderD[-(a - 1) - 1/2, z] + ParabolicCylinderD[-(a - 1) - 1/2, -(z)]) == 0</syntaxhighlight> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.0, 0.0], Times[Complex[-7.067899292141149*^-17, 0.0], GAMMA[-2.0]], Times[Complex[0.0, -8.834874115176436*^-18], GAMMA[0.0]]]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[-1.4135798584282297*^-16, 0.0], GAMMA[-2.0]], Times[Complex[-2.8271597168564594*^-16, 7.067899292141149*^-17], GAMMA[-1.0]], Times[Complex[-7.067899292141149*^-17, -8.834874115176436*^-18], GAMMA[0.0]]]
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| Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.8.E9 12.8.E9] || [[Item:Q4146|<math>\deriv[m]{}{z}\left(e^{\frac{1}{4}z^{2}}\paraU@{a}{z}\right) = (-1)^{m}\Pochhammersym{\tfrac{1}{2}+a}{m}e^{\frac{1}{4}z^{2}}\paraU@{a+m}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[m]{}{z}\left(e^{\frac{1}{4}z^{2}}\paraU@{a}{z}\right) = (-1)^{m}\Pochhammersym{\tfrac{1}{2}+a}{m}e^{\frac{1}{4}z^{2}}\paraU@{a+m}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(exp((1)/(4)*(z)^(2))*CylinderU(a, z), [z$(m)]) = (- 1)^(m)* pochhammer((1)/(2)+ a, m)*exp((1)/(4)*(z)^(2))*CylinderU(a + m, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Exp[Divide[1,4]*(z)^(2)]*ParabolicCylinderD[- 1/2 -(a), z], {z, m}] == (- 1)^(m)* Pochhammer[Divide[1,2]+ a, m]*Exp[Divide[1,4]*(z)^(2)]*ParabolicCylinderD[- 1/2 -(a + m), z]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [96 / 126]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-1.0, 0.0], DifferenceRoot[Function[{, }
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| Test Values: {Equal[Plus[Times[Plus[-1, Times[-2, ], Times[-2, -1.5]], []], Times[-2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], Plus[Times[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][1.0]], {Rule[a, -1.5], Rule[m, 1], Rule[z, Power[E, Times<syntaxhighlight lang=mathematica>Result: Plus[Complex[0.0, 0.0], Times[2.0, DifferenceRoot[Function[{, }
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| Test Values: {Equal[Plus[Times[Plus[-1, Times[-2, ], Times[-2, -1.5]], []], Times[-2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], Plus[Times[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][2.0]]], {Rule[a, -1.5], Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.8.E10 12.8.E10] || [[Item:Q4147|<math>\deriv[m]{}{z}\left(e^{-\frac{1}{4}z^{2}}\paraU@{a}{z}\right) = (-1)^{m}e^{-\frac{1}{4}z^{2}}\paraU@{a-m}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[m]{}{z}\left(e^{-\frac{1}{4}z^{2}}\paraU@{a}{z}\right) = (-1)^{m}e^{-\frac{1}{4}z^{2}}\paraU@{a-m}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(exp(-(1)/(4)*(z)^(2))*CylinderU(a, z), [z$(m)]) = (- 1)^(m)* exp(-(1)/(4)*(z)^(2))*CylinderU(a - m, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Exp[-Divide[1,4]*(z)^(2)]*ParabolicCylinderD[- 1/2 -(a), z], {z, m}] == (- 1)^(m)* Exp[-Divide[1,4]*(z)^(2)]*ParabolicCylinderD[- 1/2 -(a - m), z]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [96 / 126]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.07045205979755337, 0.7756076114781977], DifferenceRoot[Function[{, }
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| Test Values: {Equal[Plus[Times[Plus[1, Times[2, ], Times[-2, -1.5]], []], Times[2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[-1, Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]}]][1.0]], {Rule[a, -1.5], Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[2.000032302229117, -0.49556574541480647], Times[2.0, DifferenceRoot[Function[{, }
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| Test Values: {Equal[Plus[Times[Plus[1, Times[2, ], Times[-2, -1.5]], []], Times[2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[-1, Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]}]][2.0]]], {Rule[a, -1.5], Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.8.E11 12.8.E11] || [[Item:Q4148|<math>\deriv[m]{}{z}\left(e^{\frac{1}{4}z^{2}}\paraV@{a}{z}\right) = e^{\frac{1}{4}z^{2}}\paraV@{a+m}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[m]{}{z}\left(e^{\frac{1}{4}z^{2}}\paraV@{a}{z}\right) = e^{\frac{1}{4}z^{2}}\paraV@{a+m}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(exp((1)/(4)*(z)^(2))*CylinderV(a, z), [z$(m)]) = exp((1)/(4)*(z)^(2))*CylinderV(a + m, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Exp[Divide[1,4]*(z)^(2)]*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]), {z, m}] == Exp[Divide[1,4]*(z)^(2)]*Divide[GAMMA[1/2 + a + m], Pi]*(Sin[Pi*(a + m)] * ParabolicCylinderD[-(a + m) - 1/2, z] + ParabolicCylinderD[-(a + m) - 1/2, -(z)])</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [126 / 126]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[2.150599663294456*^-18, -9.777500999643939*^-18], GAMMA[0.0]], Times[0.3183098861837907, GAMMA[-1.0], Plus[DifferenceRoot[Function[{, }
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| Test Values: {Equal[Plus[Times[Plus[-1, Times[-2, ], Times[-2, -1.5]], []], Times[-2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]], Equal[[1], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], Plus[Times[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], <syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[-0.9299481905237211, -0.4298894311242862], GAMMA[1.0]], Times[0.6366197723675814, GAMMA[-1.0], Plus[DifferenceRoot[Function[{, }
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| Test Values: {Equal[Plus[Times[Plus[-1, Times[-2, ], Times[-2, -1.5]], []], Times[-2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]], Equal[[1], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], Plus[Times[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][2.0], Times[1.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[-1, Times[-2, ], Times[-2, -1.5]], []], Times[-2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[Power[E, Times[Rational[1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], Plus[Times[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[-1, ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]]}]][2.0]]]]], {Rule[a, -1.5], Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.8.E12 12.8.E12] || [[Item:Q4149|<math>\deriv[m]{}{z}\left(e^{-\frac{1}{4}z^{2}}\paraV@{a}{z}\right) = (-1)^{m}\Pochhammersym{\tfrac{1}{2}-a}{m}e^{-\frac{1}{4}z^{2}}\paraV@{a-m}{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[m]{}{z}\left(e^{-\frac{1}{4}z^{2}}\paraV@{a}{z}\right) = (-1)^{m}\Pochhammersym{\tfrac{1}{2}-a}{m}e^{-\frac{1}{4}z^{2}}\paraV@{a-m}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(exp(-(1)/(4)*(z)^(2))*CylinderV(a, z), [z$(m)]) = (- 1)^(m)* pochhammer((1)/(2)- a, m)*exp(-(1)/(4)*(z)^(2))*CylinderV(a - m, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Exp[-Divide[1,4]*(z)^(2)]*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]), {z, m}] == (- 1)^(m)* Pochhammer[Divide[1,2]- a, m]*Exp[-Divide[1,4]*(z)^(2)]*Divide[GAMMA[1/2 + a - m], Pi]*(Sin[Pi*(a - m)] * ParabolicCylinderD[-(a - m) - 1/2, z] + ParabolicCylinderD[-(a - m) - 1/2, -(z)])</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [126 / 126]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[-6.091780348003315*^-17, 1.3399109614774574*^-17], GAMMA[-2.0]], Times[0.3183098861837907, GAMMA[-1.0], Plus[DifferenceRoot[Function[{, }
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| Test Values: {Equal[Plus[Times[Plus[1, Times[2, ], Times[-2, -1.5]], []], Times[2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]], Equal[[1], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]}]][1.0], Times[1.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[1, Times[2, ], Times[-2, -1.5]], []], Ti<syntaxhighlight lang=mathematica>Result: Plus[Times[Complex[1.6052302675286988*^-15, 3.2948039393826443*^-16], GAMMA[-3.0]], Times[0.6366197723675814, GAMMA[-1.0], Plus[DifferenceRoot[Function[{, }
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| Test Values: {Equal[Plus[Times[Plus[1, Times[2, ], Times[-2, -1.5]], []], Times[2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]], Equal[[1], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Times[-1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]}]][2.0], Times[1.0, DifferenceRoot[Function[{, }, {Equal[Plus[Times[Plus[1, Times[2, ], Times[-2, -1.5]], []], Times[2, Plus[1, ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], [Plus[1, ]]], Times[2, Plus[1, ], Plus[2, ], [Plus[2, ]]]], 0], Equal[[0], Times[Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[-1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[1], Times[-1, Power[E, Times[Rational[-1, 4], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], ParabolicCylinderD[Plus[Rational[1, 2], Times[-1, -1.5]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]}]][2.0]]]]], {Rule[a, -1.5], Rule[m, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/12.10#Ex1 12.10#Ex1] || [[Item:Q4154|<math>a = +\tfrac{1}{2}\mu^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a = +\tfrac{1}{2}\mu^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a = +(1)/(2)*(mu)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a == +Divide[1,2]*\[Mu]^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/12.10#Ex2 12.10#Ex2] || [[Item:Q4155|<math>x = \mu t\sqrt{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>x = \mu t\sqrt{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">x = mu*t*sqrt(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">x == \[Mu]*t*Sqrt[2]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |-
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| | [https://dlmf.nist.gov/12.10.E2 12.10.E2] || [[Item:Q4156|<math>\deriv[2]{w}{t} = \mu^{4}(t^{2}+ 1)w</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{w}{t} = \mu^{4}(t^{2}+ 1)w</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [t$(2)]) = (mu)^(4)*((t)^(2)+ 1)*w</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {t, 2}] == \[Mu]^(4)*((t)^(2)+ 1)*w</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 2.814582564-1.625000003*I
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| Test Values: {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.625000003+2.814582564*I
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| Test Values: {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[2.814582562299425, -1.6250000000000009]
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| Test Values: {Rule[t, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[2.814582562299425, -1.6250000000000009]
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| Test Values: {Rule[t, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.10.E2 12.10.E2] || [[Item:Q4156|<math>\deriv[2]{w}{t} = \mu^{4}(t^{2}- 1)w</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{w}{t} = \mu^{4}(t^{2}- 1)w</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [t$(2)]) = (mu)^(4)*((t)^(2)- 1)*w</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {t, 2}] == \[Mu]^(4)*((t)^(2)- 1)*w</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.082531755-.6250000011*I
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| Test Values: {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .6250000011+1.082531755*I
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| Test Values: {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.0825317547305482, -0.6250000000000002]
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| Test Values: {Rule[t, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[1.0825317547305482, -0.6250000000000002]
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| Test Values: {Rule[t, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/12.10.E13 12.10.E13] || [[Item:Q4171|<math>v_{s}(t) = u_{s}(t)+\tfrac{1}{2}tu_{s-1}(t)-r_{s-2}(t)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>v_{s}(t) = u_{s}(t)+\tfrac{1}{2}tu_{s-1}(t)-r_{s-2}(t)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">v[s](t) = u[s](t)+(1)/(2)*tu[s - 1](t)- r[s - 2](t)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[v, s][t] == Subscript[u, s][t]+Divide[1,2]*Subscript[tu, s - 1][t]- Subscript[r, s - 2][t]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/12.10#Ex9 12.10#Ex9] || [[Item:Q4175|<math>\gamma_{0} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{0} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">gamma[0] = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Gamma], 0] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/12.10#Ex10 12.10#Ex10] || [[Item:Q4176|<math>\gamma_{1} = -\tfrac{1}{24}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{1} = -\tfrac{1}{24}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">gamma[1] = -(1)/(24)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Gamma], 1] == -Divide[1,24]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/12.10#Ex11 12.10#Ex11] || [[Item:Q4177|<math>\gamma_{2} = \tfrac{1}{1152}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{2} = \tfrac{1}{1152}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">gamma[2] = (1)/(1152)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Gamma], 2] == Divide[1,1152]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/12.10#Ex12 12.10#Ex12] || [[Item:Q4178|<math>\gamma_{3} = \tfrac{1003}{4\;14720}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{3} = \tfrac{1003}{4\;14720}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">gamma[3] = (1003)/(414720)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Gamma], 3] == Divide[1003,414720]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/12.10#Ex13 12.10#Ex13] || [[Item:Q4179|<math>\gamma_{4} = -\tfrac{4027}{398\;13120}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{4} = -\tfrac{4027}{398\;13120}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">gamma[4] = -(4027)/(39813120)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Gamma], 4] == -Divide[4027,39813120]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/12.10#Ex18 12.10#Ex18] || [[Item:Q4198|<math>\mathsf{A}_{1}(\tau) = -\tfrac{1}{12}\tau(20\tau^{2}+30\tau+9)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathsf{A}_{1}(\tau) = -\tfrac{1}{12}\tau(20\tau^{2}+30\tau+9)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">A[1]*(((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))) = -(1)/(12)*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))*(20*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))^(2)+ 30*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))+ 9)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[A, 1]*((Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))) == -Divide[1,12]*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))*(20*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))^(2)+ 30*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))+ 9)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/12.10#Ex19 12.10#Ex19] || [[Item:Q4199|<math>\mathsf{A}_{2}(\tau) = \tfrac{1}{288}\tau^{2}(6160\tau^{4}+18480\tau^{3}+19404\tau^{2}+8028\tau+945)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathsf{A}_{2}(\tau) = \tfrac{1}{288}\tau^{2}(6160\tau^{4}+18480\tau^{3}+19404\tau^{2}+8028\tau+945)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">A[2]*(((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))) = (1)/(288)*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))^(2)*(6160*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))^(4)+ 18480*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))^(3)+ 19404*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))^(2)+ 8028*((1)/(2)*((t)/(sqrt((t)^(2)- 1))- 1))+ 945)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[A, 2]*((Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))) == Divide[1,288]*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))^(2)*(6160*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))^(4)+ 18480*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))^(3)+ 19404*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))^(2)+ 8028*(Divide[1,2]*(Divide[t,Sqrt[(t)^(2)- 1]]- 1))+ 945)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/12.10#Ex22 12.10#Ex22] || [[Item:Q4208|<math>A_{s}(\zeta) = \zeta^{-3s}\sum_{m=0}^{2s}\beta_{m}(\phi(\zeta))^{6(2s-m)}u_{2s-m}(t)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>A_{s}(\zeta) = \zeta^{-3s}\sum_{m=0}^{2s}\beta_{m}(\phi(\zeta))^{6(2s-m)}u_{2s-m}(t)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">A[s](zeta) = (zeta)^(- 3*s)* sum((-(6*m + 1)/(6*m - 1)*alpha[m])*((((zeta)/((t)^(2)- 1))^((1)/(4))))^(6*(2*s - m))* u[2*s - m](t), m = 0..2*s)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[A, s][\[Zeta]] == \[Zeta]^(- 3*s)* Sum[(-Divide[6*m + 1,6*m - 1]*Subscript[\[Alpha], m])*(((Divide[\[Zeta],(t)^(2)- 1])^(Divide[1,4])))^(6*(2*s - m))* Subscript[u, 2*s - m][t], {m, 0, 2*s}, GenerateConditions->None]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/12.10#Ex23 12.10#Ex23] || [[Item:Q4209|<math>\zeta^{2}B_{s}(\zeta) = -\zeta^{-3s}\sum_{m=0}^{2s+1}\alpha_{m}(\phi(\zeta))^{6(2s-m+1)}u_{2s-m+1}(t)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\zeta^{2}B_{s}(\zeta) = -\zeta^{-3s}\sum_{m=0}^{2s+1}\alpha_{m}(\phi(\zeta))^{6(2s-m+1)}u_{2s-m+1}(t)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(zeta)^(2)* B[s](zeta) = - (zeta)^(- 3*s)* sum((alpha[m]((((zeta)/((t)^(2)- 1))^((1)/(4)))))^(6*(2*s - m + 1))* u[2*s - m + 1](t), m = 0..2*s + 1)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">\[Zeta]^(2)* Subscript[B, s][\[Zeta]] == - \[Zeta]^(- 3*s)* Sum[(Subscript[\[Alpha], m][((Divide[\[Zeta],(t)^(2)- 1])^(Divide[1,4]))])^(6*(2*s - m + 1))* Subscript[u, 2*s - m + 1][t], {m, 0, 2*s + 1}, GenerateConditions->None]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/12.10#Ex28 12.10#Ex28] || [[Item:Q4215|<math>\zeta C_{s}(\zeta) = -\zeta^{-3s}\sum_{m=0}^{2s+1}\beta_{m}(\phi(\zeta))^{6(2s-m+1)}v_{2s-m+1}(t)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\zeta C_{s}(\zeta) = -\zeta^{-3s}\sum_{m=0}^{2s+1}\beta_{m}(\phi(\zeta))^{6(2s-m+1)}v_{2s-m+1}(t)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">zeta*C[s](zeta) = - (zeta)^(- 3*s)* sum((-(6*m + 1)/(6*m - 1)*alpha[m])*((((zeta)/((t)^(2)- 1))^((1)/(4))))^(6*(2*s - m + 1))* v[2*s - m + 1](t), m = 0..2*s + 1)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">\[Zeta]*Subscript[C, s][\[Zeta]] == - \[Zeta]^(- 3*s)* Sum[(-Divide[6*m + 1,6*m - 1]*Subscript[\[Alpha], m])*(((Divide[\[Zeta],(t)^(2)- 1])^(Divide[1,4])))^(6*(2*s - m + 1))* Subscript[v, 2*s - m + 1][t], {m, 0, 2*s + 1}, GenerateConditions->None]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/12.10#Ex29 12.10#Ex29] || [[Item:Q4216|<math>D_{s}(\zeta) = \zeta^{-3s}\sum_{m=0}^{2s}\alpha_{m}(\phi(\zeta))^{6(2s-m)}v_{2s-m}(t)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>D_{s}(\zeta) = \zeta^{-3s}\sum_{m=0}^{2s}\alpha_{m}(\phi(\zeta))^{6(2s-m)}v_{2s-m}(t)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">D[s](zeta) = (zeta)^(- 3*s)* sum((alpha[m]((((zeta)/((t)^(2)- 1))^((1)/(4)))))^(6*(2*s - m))* v[2*s - m](t), m = 0..2*s)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[D, s][\[Zeta]] == \[Zeta]^(- 3*s)* Sum[(Subscript[\[Alpha], m][((Divide[\[Zeta],(t)^(2)- 1])^(Divide[1,4]))])^(6*(2*s - m))* Subscript[v, 2*s - m][t], {m, 0, 2*s}, GenerateConditions->None]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/12.11.E5 12.11.E5] || [[Item:Q4221|<math>p_{0}(\zeta) = t(\zeta)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>p_{0}(\zeta) = t(\zeta)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">p[0](zeta) = t(zeta)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[p, 0][\[Zeta]] == t[\[Zeta]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/12.11.E6 12.11.E6] || [[Item:Q4222|<math>p_{1}(\zeta) = \frac{t^{3}-6t}{24(t^{2}-1)^{2}}+\frac{5}{48((t^{2}-1)\zeta^{3})^{\frac{1}{2}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>p_{1}(\zeta) = \frac{t^{3}-6t}{24(t^{2}-1)^{2}}+\frac{5}{48((t^{2}-1)\zeta^{3})^{\frac{1}{2}}}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">p[1](zeta) = ((t)^(3)- 6*t)/(24*((t)^(2)- 1)^(2))+(5)/(48*(((t)^(2)- 1)*(zeta)^(3))^((1)/(2)))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[p, 1][\[Zeta]] == Divide[(t)^(3)- 6*t,24*((t)^(2)- 1)^(2)]+Divide[5,48*(((t)^(2)- 1)*\[Zeta]^(3))^(Divide[1,2])]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |- style="background: #dfe6e9;"
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| | [https://dlmf.nist.gov/12.11.E8 12.11.E8] || [[Item:Q4224|<math>q_{0}(\zeta) = t(\zeta)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>q_{0}(\zeta) = t(\zeta)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">q[0](zeta) = t(zeta)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[q, 0][\[Zeta]] == t[\[Zeta]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/12.12.E1 12.12.E1] || [[Item:Q4226|<math>\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{\mu-1}\paraU@{a}{t}\diff{t} = \frac{\sqrt{\pi}2^{-\frac{1}{2}(\mu+a+\frac{1}{2})}\EulerGamma@{\mu}}{\EulerGamma@{\frac{1}{2}(\mu+a+\frac{3}{2})}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{\mu-1}\paraU@{a}{t}\diff{t} = \frac{\sqrt{\pi}2^{-\frac{1}{2}(\mu+a+\frac{1}{2})}\EulerGamma@{\mu}}{\EulerGamma@{\frac{1}{2}(\mu+a+\frac{3}{2})}}</syntaxhighlight> || <math>\realpart@@{\mu} > 0, \realpart@@{(\mu)} > 0, \realpart@@{(\frac{1}{2}(\mu+a+\frac{3}{2}))} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(-(1)/(4)*(t)^(2))*(t)^(mu - 1)* CylinderU(a, t), t = 0..infinity) = (sqrt(Pi)*(2)^(-(1)/(2)*(mu + a +(1)/(2)))* GAMMA(mu))/(GAMMA((1)/(2)*(mu + a +(3)/(2))))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[-Divide[1,4]*(t)^(2)]*(t)^(\[Mu]- 1)* ParabolicCylinderD[- 1/2 -(a), t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Sqrt[Pi]*(2)^(-Divide[1,2]*(\[Mu]+ a +Divide[1,2]))* Gamma[\[Mu]],Gamma[Divide[1,2]*(\[Mu]+ a +Divide[3,2])]]</syntaxhighlight> || Successful || Aborted || - || Skipped - Because timed out
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| | [https://dlmf.nist.gov/12.12.E2 12.12.E2] || [[Item:Q4227|<math>\int_{0}^{\infty}e^{-\frac{3}{4}t^{2}}t^{-a-\frac{3}{2}}\paraU@{a}{t}\diff{t} = 2^{\frac{1}{4}+\frac{1}{2}a}\EulerGamma@{-a-\tfrac{1}{2}}\cos@{(\tfrac{1}{4}a+\tfrac{1}{8})\pi}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-\frac{3}{4}t^{2}}t^{-a-\frac{3}{2}}\paraU@{a}{t}\diff{t} = 2^{\frac{1}{4}+\frac{1}{2}a}\EulerGamma@{-a-\tfrac{1}{2}}\cos@{(\tfrac{1}{4}a+\tfrac{1}{8})\pi}</syntaxhighlight> || <math>\realpart@@{a} < -\tfrac{1}{2}, \realpart@@{(-a-\tfrac{1}{2})} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(-(3)/(4)*(t)^(2))*(t)^(- a -(3)/(2))* CylinderU(a, t), t = 0..infinity) = (2)^((1)/(4)+(1)/(2)*a)* GAMMA(- a -(1)/(2))*cos(((1)/(4)*a +(1)/(8))*Pi)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[-Divide[3,4]*(t)^(2)]*(t)^(- a -Divide[3,2])* ParabolicCylinderD[- 1/2 -(a), t], {t, 0, Infinity}, GenerateConditions->None] == (2)^(Divide[1,4]+Divide[1,2]*a)* Gamma[- a -Divide[1,2]]*Cos[(Divide[1,4]*a +Divide[1,8])*Pi]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Successful [Tested: 2]
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| | [https://dlmf.nist.gov/12.12.E3 12.12.E3] || [[Item:Q4228|<math>\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{-a-\frac{1}{2}}(x^{2}+t^{2})^{-1}\paraU@{a}{t}\diff{t} = \sqrt{\pi/2}\EulerGamma@{\tfrac{1}{2}-a}x^{-a-\frac{3}{2}}e^{\frac{1}{4}x^{2}}\paraU@{-a}{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{-a-\frac{1}{2}}(x^{2}+t^{2})^{-1}\paraU@{a}{t}\diff{t} = \sqrt{\pi/2}\EulerGamma@{\tfrac{1}{2}-a}x^{-a-\frac{3}{2}}e^{\frac{1}{4}x^{2}}\paraU@{-a}{x}</syntaxhighlight> || <math>x > 0, \realpart@@{a} < \tfrac{1}{2}, \realpart@@{(\tfrac{1}{2}-a)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(-(1)/(4)*(t)^(2))*(t)^(- a -(1)/(2))*((x)^(2)+ (t)^(2))^(- 1)* CylinderU(a, t), t = 0..infinity) = sqrt(Pi/2)*GAMMA((1)/(2)- a)*(x)^(- a -(3)/(2))* exp((1)/(4)*(x)^(2))*CylinderU(- a, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[-Divide[1,4]*(t)^(2)]*(t)^(- a -Divide[1,2])*((x)^(2)+ (t)^(2))^(- 1)* ParabolicCylinderD[- 1/2 -(a), t], {t, 0, Infinity}, GenerateConditions->None] == Sqrt[Pi/2]*Gamma[Divide[1,2]- a]*(x)^(- a -Divide[3,2])* Exp[Divide[1,4]*(x)^(2)]*ParabolicCylinderD[- 1/2 -(- a), x]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/12.13.E1 12.13.E1] || [[Item:Q4230|<math>\paraU@{a}{x+y} = e^{\frac{1}{2}xy+\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\frac{(-y)^{m}}{m!}\paraU@{a-m}{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{x+y} = e^{\frac{1}{2}xy+\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\frac{(-y)^{m}}{m!}\paraU@{a-m}{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(a, x + y) = exp((1)/(2)*x*y +(1)/(4)*(y)^(2))*sum(((- y)^(m))/(factorial(m))*CylinderU(a - m, x), m = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), x + y] == Exp[Divide[1,2]*x*y +Divide[1,4]*(y)^(2)]*Sum[Divide[(- y)^(m),(m)!]*ParabolicCylinderD[- 1/2 -(a - m), x], {m, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/12.13.E2 12.13.E2] || [[Item:Q4231|<math>\paraU@{a}{x+y} = e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\binom{-a-\tfrac{1}{2}}{m}y^{m}\paraU@{a+m}{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{x+y} = e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\binom{-a-\tfrac{1}{2}}{m}y^{m}\paraU@{a+m}{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderU(a, x + y) = exp(-(1)/(2)*x*y -(1)/(4)*(y)^(2))*sum(binomial(- a -(1)/(2),m)*(y)^(m)* CylinderU(a + m, x), m = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), x + y] == Exp[-Divide[1,2]*x*y -Divide[1,4]*(y)^(2)]*Sum[Binomial[- a -Divide[1,2],m]*(y)^(m)* ParabolicCylinderD[- 1/2 -(a + m), x], {m, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/12.13.E3 12.13.E3] || [[Item:Q4232|<math>\paraV@{a}{x+y} = e^{\frac{1}{2}xy+\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\binom{a-\tfrac{1}{2}}{m}y^{m}\paraV@{a-m}{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV@{a}{x+y} = e^{\frac{1}{2}xy+\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\binom{a-\tfrac{1}{2}}{m}y^{m}\paraV@{a-m}{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderV(a, x + y) = exp((1)/(2)*x*y +(1)/(4)*(y)^(2))*sum(binomial(a -(1)/(2),m)*(y)^(m)* CylinderV(a - m, x), m = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, x + y] + ParabolicCylinderD[-(a) - 1/2, -(x + y)]) == Exp[Divide[1,2]*x*y +Divide[1,4]*(y)^(2)]*Sum[Binomial[a -Divide[1,2],m]*(y)^(m)* Divide[GAMMA[1/2 + a - m], Pi]*(Sin[Pi*(a - m)] * ParabolicCylinderD[-(a - m) - 1/2, x] + ParabolicCylinderD[-(a - m) - 1/2, -(x)]), {m, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/12.13.E4 12.13.E4] || [[Item:Q4233|<math>\paraV@{a}{x+y} = e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\frac{y^{m}}{m!}\paraV@{a+m}{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraV@{a}{x+y} = e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\frac{y^{m}}{m!}\paraV@{a+m}{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>CylinderV(a, x + y) = exp(-(1)/(2)*x*y -(1)/(4)*(y)^(2))*sum(((y)^(m))/(factorial(m))*CylinderV(a + m, x), m = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, x + y] + ParabolicCylinderD[-(a) - 1/2, -(x + y)]) == Exp[-Divide[1,2]*x*y -Divide[1,4]*(y)^(2)]*Sum[Divide[(y)^(m),(m)!]*Divide[GAMMA[1/2 + a + m], Pi]*(Sin[Pi*(a + m)] * ParabolicCylinderD[-(a + m) - 1/2, x] + ParabolicCylinderD[-(a + m) - 1/2, -(x)]), {m, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/12.13.E5 12.13.E5] || [[Item:Q4234|<math>\paraU@{a}{x\cos@@{t}+y\sin@@{t}}\\ = e^{\frac{1}{4}(x\sin@@{t}-y\cos@@{t})^{2}}\*\sum_{m=0}^{\infty}\binom{-a-\tfrac{1}{2}}{m}(\tan@@{t})^{m}\paraU@{m+a}{x}\paraU@{-m-\tfrac{1}{2}}{y}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraU@{a}{x\cos@@{t}+y\sin@@{t}}\\ = e^{\frac{1}{4}(x\sin@@{t}-y\cos@@{t})^{2}}\*\sum_{m=0}^{\infty}\binom{-a-\tfrac{1}{2}}{m}(\tan@@{t})^{m}\paraU@{m+a}{x}\paraU@{-m-\tfrac{1}{2}}{y}</syntaxhighlight> || <math>\realpart@@{a} \leq -\tfrac{1}{2}, 0 \leq t, t \leq \tfrac{1}{4}\pi</math> || <syntaxhighlight lang=mathematica>CylinderU(a, x*cos(t)+ y*sin(t)) = exp((1)/(4)*(x*sin(t)- y*cos(t))^(2))* sum(binomial(- a -(1)/(2),m)*(tan(t))^(m)* CylinderU(m + a, x)*CylinderU(- m -(1)/(2), y), m = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>ParabolicCylinderD[- 1/2 -(a), x*Cos[t]+ y*Sin[t]] == Exp[Divide[1,4]*(x*Sin[t]- y*Cos[t])^(2)]* Sum[Binomial[- a -Divide[1,2],m]*(Tan[t])^(m)* ParabolicCylinderD[- 1/2 -(m + a), x]*ParabolicCylinderD[- 1/2 -(- m -Divide[1,2]), y], {m, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skip - No test values generated
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| | [https://dlmf.nist.gov/12.14.E1 12.14.E1] || [[Item:Q4236|<math>\paraW@{a}{0} = 2^{-\frac{3}{4}}\left|\frac{\EulerGamma@{\tfrac{1}{4}+\tfrac{1}{2}ia}}{\EulerGamma@{\tfrac{3}{4}+\tfrac{1}{2}ia}}\right|^{\frac{1}{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraW@{a}{0} = 2^{-\frac{3}{4}}\left|\frac{\EulerGamma@{\tfrac{1}{4}+\tfrac{1}{2}ia}}{\EulerGamma@{\tfrac{3}{4}+\tfrac{1}{2}ia}}\right|^{\frac{1}{2}}</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{4}+\tfrac{1}{2}\iunit a)} > 0, \realpart@@{(\tfrac{3}{4}+\tfrac{1}{2}\iunit a)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[(Sqrt[1+Exp[2*Pi*(a)]]-Exp[Pi*(a)])/2] * Exp[Divide[Pi*(a),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 - I*(a), 0 * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 + I*(a), 0 * Exp[Divide[Pi*I,4]]] ) == (2)^(-Divide[3,4])*(Abs[Divide[Gamma[Divide[1,4]+Divide[1,2]*I*a],Gamma[Divide[3,4]+Divide[1,2]*I*a]]])^(Divide[1,2])</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 6]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.6502446611528931, Times[0.2167171091323973, Plus[Times[Complex[2.1101734540747557, 0.09157129889910319], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, -1.5]]]]]]]], Times[Complex[2.1101734540747557, -0.09157129889910319], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, -1.5]]]]]]]]]]]
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| Test Values: {Rule[a, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.6502446611528931, Times[0.15393043293932354, Plus[Times[Complex[2.1101734540747557, -0.09157129889910319], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, 1.5]]]]]]]], Times[Complex[2.1101734540747557, 0.09157129889910319], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, 1.5]]]]]]]]]]]
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| Test Values: {Rule[a, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.14.E2 12.14.E2] || [[Item:Q4237|<math>\paraW'@{a}{0} = -2^{-\frac{1}{4}}\left|\frac{\EulerGamma@{\tfrac{3}{4}+\tfrac{1}{2}ia}}{\EulerGamma@{\tfrac{1}{4}+\tfrac{1}{2}ia}}\right|^{\frac{1}{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraW'@{a}{0} = -2^{-\frac{1}{4}}\left|\frac{\EulerGamma@{\tfrac{3}{4}+\tfrac{1}{2}ia}}{\EulerGamma@{\tfrac{1}{4}+\tfrac{1}{2}ia}}\right|^{\frac{1}{2}}</syntaxhighlight> || <math>\realpart@@{(\tfrac{3}{4}+\tfrac{1}{2}\iunit a)} > 0, \realpart@@{(\tfrac{1}{4}+\tfrac{1}{2}\iunit a)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[Sqrt[(Sqrt[1+Exp[2*Pi*(a)]]-Exp[Pi*(a)])/2] * Exp[Divide[Pi*(a),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 - I*(a), temp * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 + I*(a), temp * Exp[Divide[Pi*I,4]]] ), {temp, 1}]/.temp-> 0) == - (2)^(-Divide[1,4])*(Abs[Divide[Gamma[Divide[3,4]+Divide[1,2]*I*a],Gamma[Divide[1,4]+Divide[1,2]*I*a]]])^(Divide[1,2])</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 6]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.7689413383471582, Times[0.2167171091323973, Plus[Times[Complex[-1.704391150531108, -1.8258251253417301], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, -1.5]]]]]]]]], Times[Complex[-1.704391150531108, 1.8258251253417301], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, -1.5]]]]]]]]]]]]
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| Test Values: {Rule[a, -1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[0.7689413383471582, Times[0.15393043293932354, Plus[Times[Complex[-1.704391150531108, 1.8258251253417301], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, 1.5]]]]]]]]], Times[Complex[-1.704391150531108, -1.8258251253417301], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, 1.5]]]]]]]]]]]]
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| Test Values: {Rule[a, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| |-
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| | [https://dlmf.nist.gov/12.14.E3 12.14.E3] || [[Item:Q4238|<math>\Wronskian@{\paraW@{a}{x},\paraW@{a}{-x}} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\paraW@{a}{x},\paraW@{a}{-x}} = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{Sqrt[(Sqrt[1+Exp[2*Pi*(a)]]-Exp[Pi*(a)])/2] * Exp[Divide[Pi*(a),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 - I*(a), x * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 + I*(a), x * Exp[Divide[Pi*I,4]]] ), Sqrt[(Sqrt[1+Exp[2*Pi*(a)]]-Exp[Pi*(a)])/2] * Exp[Divide[Pi*(a),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 - I*(a), - x * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 + I*(a), - x * Exp[Divide[Pi*I,4]]] )}, x] == 1</syntaxhighlight> || Missing Macro Error || Aborted || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-1.0, Times[0.49552852896181854, Power[2.718281828459045, Plus[-2.356194490192345, Times[Complex[0.0, -1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]]], Plus[Complex[6.858735565841029, 8.017762045530217], Times[Complex[-3.325234230733274, 7.771974729433958], Power[2.718281828459045, Times[Complex[0.0, 1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]]], Times[Complex[-0.5683445061301408, 1.832896863544324], Power[2.718281828459045, Times[Complex[0.0, 1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]], Plus[Complex[-3.829019967249232, -1.729594934825754], Times[Complex[-1.3099191255337557, -0.33304402326481836], Power[2.718281828459045, Times[Complex[0.0, 1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]]]]], Times[Complex[-0.6925599504260578, -1.7781797223294367], Power[2.718281828459045, Times[Complex[0.0, 1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]], Plus[Complex[1.1964059764236668, -0.5640566230504777], Times[Complex[1.7883988364886165, 3.791736125757196], Power[2.718281828459045, Times<syntaxhighlight lang=mathematica>Result: Plus[-1.0, Times[0.49552852896181854, Power[2.718281828459045, Plus[-2.356194490192345, Times[Complex[0.0, -1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]]], Plus[Complex[6.858735565841018, 8.01776204553021], Times[Complex[2.1162015796075306, -3.4943658291294586], Power[2.718281828459045, Times[Complex[0.0, 1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]]], Times[Complex[1.7644828790311722, 1.0958018333501354], Power[2.718281828459045, Times[Complex[0.0, 1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]], Plus[Complex[-1.886026428459961, 2.3891008291554687], Times[Complex[0.20069396529457476, -1.987274101858182], Power[2.718281828459045, Times[Complex[0.0, 1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]]]]], Times[Complex[1.683947543704261, -1.245446997525291], Power[2.718281828459045, Times[Complex[0.0, 1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]], Plus[Complex[1.1608218100872771, 1.6575881988304835], Times[Complex[3.0041801875943053, -0.5876813330718609], Power[2.718281828459045, Times[Complex[0.0, 1.0], Arg[GAMMA[Complex[0.5, -1.5]]]]]]]]]]]
| |
| Test Values: {Rule[a, -1.5], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
| |
| |-
| |
| | [https://dlmf.nist.gov/12.14.E4 12.14.E4] || [[Item:Q4239|<math>\paraW@{a}{x} = \sqrt{k/2}\,e^{\frac{1}{4}\pi a}\left(e^{i\rho}\paraU@{ia}{xe^{-\pi i/4}}+e^{-i\rho}\paraU@{-ia}{xe^{\pi i/4}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraW@{a}{x} = \sqrt{k/2}\,e^{\frac{1}{4}\pi a}\left(e^{i\rho}\paraU@{ia}{xe^{-\pi i/4}}+e^{-i\rho}\paraU@{-ia}{xe^{\pi i/4}}\right)</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}+\iunit a)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[(Sqrt[1+Exp[2*Pi*(a)]]-Exp[Pi*(a)])/2] * Exp[Divide[Pi*(a),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 - I*(a), x * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 + I*(a), x * Exp[Divide[Pi*I,4]]] ) == Sqrt[(Sqrt[1 + Exp[2*Pi*a]]- Exp[Pi*a])/2]*Exp[Divide[1,4]*Pi*a]*(Exp[I*(Divide[1,8]*Pi +Divide[1,2]*(Arg[Gamma[Divide[1,2]+ I*a]]))]*ParabolicCylinderD[- 1/2 -(I*a), x*Exp[- Pi*I/4]]+ Exp[- I*(Divide[1,8]*Pi +Divide[1,2]*(Arg[Gamma[Divide[1,2]+ I*a]]))]*ParabolicCylinderD[- 1/2 -(- I*a), x*Exp[Pi*I/4]])</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.7504500073451766, 0.0], Times[0.2167171091323973, Plus[Times[Complex[-0.5683445061301404, -1.832896863544323], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, -1.5]]]]]]]], Times[Complex[-0.5683445061301404, 1.832896863544323], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, -1.5]]]]]]]]]]]
| |
| Test Values: {Rule[a, -1.5], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.17071363418721158, 0.0], Times[0.2167171091323973, Plus[Times[Complex[1.764482879031172, -1.0958018333501351], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, -1.5]]]]]]]], Times[Complex[1.764482879031172, 1.0958018333501351], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[Complex[0.5, -1.5]]]]]]]]]]]
| |
| Test Values: {Rule[a, -1.5], Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
| |
| |-
| |
| | [https://dlmf.nist.gov/12.14.E8 12.14.E8] || [[Item:Q4244|<math>\paraW@{a}{x} = \paraW@{a}{0}w_{1}(a,x)+\paraW'@{a}{0}w_{2}(a,x)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraW@{a}{x} = \paraW@{a}{0}w_{1}(a,x)+\paraW'@{a}{0}w_{2}(a,x)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[(Sqrt[1+Exp[2*Pi*(a)]]-Exp[Pi*(a)])/2] * Exp[Divide[Pi*(a),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 - I*(a), x * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 + I*(a), x * Exp[Divide[Pi*I,4]]] ) == Sqrt[(Sqrt[1+Exp[2*Pi*(a)]]-Exp[Pi*(a)])/2] * Exp[Divide[Pi*(a),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 - I*(a), 0 * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 + I*(a), 0 * Exp[Divide[Pi*I,4]]] )*(Sum[Subscript[\[Alpha], n][a]*Divide[(x)^(2*n),(2*n)!], {n, 0, Infinity}, GenerateConditions->None])+ (D[Sqrt[(Sqrt[1+Exp[2*Pi*(a)]]-Exp[Pi*(a)])/2] * Exp[Divide[Pi*(a),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 - I*(a), temp * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 + I*(a), temp * Exp[Divide[Pi*I,4]]] ), {temp, 1}]/.temp-> 0)*(Sum[Subscript[\[Beta], n][a]*Divide[(x)^(2*n + 1),(2*n + 1)!], {n, 0, Infinity}, GenerateConditions->None])</syntaxhighlight> || Missing Macro Error || Aborted || - || Skipped - Because timed out
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/12.14#Ex5 12.14#Ex5] || [[Item:Q4249|<math>\alpha_{0}(a) = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\alpha_{0}(a) = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">alpha[0](a) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Alpha], 0][a] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/12.14#Ex6 12.14#Ex6] || [[Item:Q4250|<math>\alpha_{1}(a) = a</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\alpha_{1}(a) = a</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">alpha[1](a) = a</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Alpha], 1][a] == a</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/12.14#Ex7 12.14#Ex7] || [[Item:Q4251|<math>\beta_{0}(a) = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\beta_{0}(a) = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">beta[0](a) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Beta], 0][a] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| |
| |- style="background: #dfe6e9;"
| |
| | [https://dlmf.nist.gov/12.14#Ex8 12.14#Ex8] || [[Item:Q4252|<math>\beta_{1}(a) = a</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\beta_{1}(a) = a</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">beta[1](a) = a</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Beta], 1][a] == a</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| |-
| |
| | [https://dlmf.nist.gov/12.14.E13 12.14.E13] || [[Item:Q4253|<math>\paraW@{0}{+ x} = 2^{-\frac{5}{4}}\sqrt{\pi x}\left(\BesselJ{-\frac{1}{4}}@{\tfrac{1}{4}x^{2}}-\BesselJ{\frac{1}{4}}@{\tfrac{1}{4}x^{2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraW@{0}{+ x} = 2^{-\frac{5}{4}}\sqrt{\pi x}\left(\BesselJ{-\frac{1}{4}}@{\tfrac{1}{4}x^{2}}-\BesselJ{\frac{1}{4}}@{\tfrac{1}{4}x^{2}}\right)</syntaxhighlight> || <math>\realpart@@{((-\frac{1}{4})+k+1)} > 0, \realpart@@{((\frac{1}{4})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[(Sqrt[1+Exp[2*Pi*(0)]]-Exp[Pi*(0)])/2] * Exp[Divide[Pi*(0),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 - I*(0), + x * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 + I*(0), + x * Exp[Divide[Pi*I,4]]] ) == (2)^(-Divide[5,4])*Sqrt[Pi*x]*(BesselJ[-Divide[1,4], Divide[1,4]*(x)^(2)]- BesselJ[Divide[1,4], Divide[1,4]*(x)^(2)])</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.22960009916312846, Times[0.4550898605622274, Plus[Times[Complex[0.5125789656744846, -0.578293218532047], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[0.5125789656744846, 0.578293218532047], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]
| |
| Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.7771899742615831, Times[0.4550898605622274, Plus[Times[Complex[1.0093127652068992, -0.20538419268274744], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[1.0093127652068992, 0.20538419268274744], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]
| |
| Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
| |
| |-
| |
| | [https://dlmf.nist.gov/12.14.E13 12.14.E13] || [[Item:Q4253|<math>\paraW@{0}{- x} = 2^{-\frac{5}{4}}\sqrt{\pi x}\left(\BesselJ{-\frac{1}{4}}@{\tfrac{1}{4}x^{2}}+\BesselJ{\frac{1}{4}}@{\tfrac{1}{4}x^{2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraW@{0}{- x} = 2^{-\frac{5}{4}}\sqrt{\pi x}\left(\BesselJ{-\frac{1}{4}}@{\tfrac{1}{4}x^{2}}+\BesselJ{\frac{1}{4}}@{\tfrac{1}{4}x^{2}}\right)</syntaxhighlight> || <math>\realpart@@{((-\frac{1}{4})+k+1)} > 0, \realpart@@{((\frac{1}{4})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[(Sqrt[1+Exp[2*Pi*(0)]]-Exp[Pi*(0)])/2] * Exp[Divide[Pi*(0),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 - I*(0), - x * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 + I*(0), - x * Exp[Divide[Pi*I,4]]] ) == (2)^(-Divide[5,4])*Sqrt[Pi*x]*(BesselJ[-Divide[1,4], Divide[1,4]*(x)^(2)]+ BesselJ[Divide[1,4], Divide[1,4]*(x)^(2)])</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-1.6050209192353964, Times[0.4550898605622274, Plus[Times[Complex[1.669165402738578, 0.5782932185320475], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[1.669165402738578, -0.5782932185320475], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]
| |
| Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-1.2656786607564097, Times[0.4550898605622274, Plus[Times[Complex[1.4200811505723943, 0.2053841926827476], Power[2.718281828459045, Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]], Times[Complex[1.4200811505723943, -0.2053841926827476], Power[2.718281828459045, Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]
| |
| Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| |-
| |
| | [https://dlmf.nist.gov/12.14.E14 12.14.E14] || [[Item:Q4254|<math>\deriv{}{x}\paraW@{0}{+ x} = -2^{-\frac{9}{4}}x\sqrt{\pi x}\left(\BesselJ{\frac{3}{4}}@{\tfrac{1}{4}x^{2}}+\BesselJ{-\frac{3}{4}}@{\tfrac{1}{4}x^{2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv{}{x}\paraW@{0}{+ x} = -2^{-\frac{9}{4}}x\sqrt{\pi x}\left(\BesselJ{\frac{3}{4}}@{\tfrac{1}{4}x^{2}}+\BesselJ{-\frac{3}{4}}@{\tfrac{1}{4}x^{2}}\right)</syntaxhighlight> || <math>\realpart@@{((\frac{3}{4})+k+1)} > 0, \realpart@@{((-\frac{3}{4})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Sqrt[(Sqrt[1+Exp[2*Pi*(0)]]-Exp[Pi*(0)])/2] * Exp[Divide[Pi*(0),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 - I*(0), + x * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 + I*(0), + x * Exp[Divide[Pi*I,4]]] ), x] == - (2)^(-Divide[9,4])* x*Sqrt[Pi*x]*(BesselJ[Divide[3,4], Divide[1,4]*(x)^(2)]+ BesselJ[-Divide[3,4], Divide[1,4]*(x)^(2)])</syntaxhighlight> || Missing Macro Error || Aborted || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.6138624292597322, Times[0.4550898605622274, Plus[Times[Complex[-0.6342811205261311, 0.23110891742402956], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[-0.6342811205261311, -0.23110891742402956], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]]
| |
| Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[0.497609493984496, Times[0.4550898605622274, Plus[Times[Complex[-0.5880519854532475, 0.008953751453165265], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[-0.5880519854532475, -0.008953751453165265], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]]
| |
| Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.14.E14 12.14.E14] || [[Item:Q4254|<math>\deriv{}{x}\paraW@{0}{- x} = -2^{-\frac{9}{4}}x\sqrt{\pi x}\left(\BesselJ{\frac{3}{4}}@{\tfrac{1}{4}x^{2}}-\BesselJ{-\frac{3}{4}}@{\tfrac{1}{4}x^{2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv{}{x}\paraW@{0}{- x} = -2^{-\frac{9}{4}}x\sqrt{\pi x}\left(\BesselJ{\frac{3}{4}}@{\tfrac{1}{4}x^{2}}-\BesselJ{-\frac{3}{4}}@{\tfrac{1}{4}x^{2}}\right)</syntaxhighlight> || <math>\realpart@@{((\frac{3}{4})+k+1)} > 0, \realpart@@{((-\frac{3}{4})+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[Sqrt[(Sqrt[1+Exp[2*Pi*(0)]]-Exp[Pi*(0)])/2] * Exp[Divide[Pi*(0),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 - I*(0), - x * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(0)]]/2)] * ParabolicCylinderD[- 1/2 + I*(0), - x * Exp[Divide[Pi*I,4]]] ), x] == - (2)^(-Divide[9,4])* x*Sqrt[Pi*x]*(BesselJ[Divide[3,4], Divide[1,4]*(x)^(2)]- BesselJ[-Divide[3,4], Divide[1,4]*(x)^(2)])</syntaxhighlight> || Missing Macro Error || Aborted || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.06418969137726768, Times[0.4550898605622274, Plus[Times[Complex[0.17206328567807166, 0.23110891742402973], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[0.17206328567807166, -0.23110891742402973], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]]
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| Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.4763137641163559, Times[0.4550898605622274, Plus[Times[Complex[0.5701444825469169, 0.008953751453165182], Power[2.718281828459045, Plus[Complex[0.0, 0.7853981633974483], Times[Complex[0.0, -1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]], Times[Complex[0.5701444825469169, -0.008953751453165182], Power[2.718281828459045, Plus[Complex[0.0, -0.7853981633974483], Times[Complex[0.0, 1.0], Plus[0.39269908169872414, Times[0.5, Arg[GAMMA[0.5]]]]]]]]]]]
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| Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.14.E15 12.14.E15] || [[Item:Q4255|<math>w_{1}(a,x) = e^{-\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{1}{4}-\tfrac{1}{2}ia}{\tfrac{1}{2}}{\tfrac{1}{2}ix^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>w_{1}(a,x) = e^{-\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{1}{4}-\tfrac{1}{2}ia}{\tfrac{1}{2}}{\tfrac{1}{2}ix^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(sum(alpha[n](a)*((x)^(2*n))/(factorial(2*n)), n = 0..infinity)) = exp(-(1)/(4)*I*(x)^(2))*KummerM((1)/(4)-(1)/(2)*I*a, (1)/(2), (1)/(2)*I*(x)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Sum[Subscript[\[Alpha], n][a]*Divide[(x)^(2*n),(2*n)!], {n, 0, Infinity}, GenerateConditions->None]) == Exp[-Divide[1,4]*I*(x)^(2)]*Hypergeometric1F1[Divide[1,4]-Divide[1,2]*I*a, Divide[1,2], Divide[1,2]*I*(x)^(2)]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/12.14.E15 12.14.E15] || [[Item:Q4255|<math>e^{-\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{1}{4}-\tfrac{1}{2}ia}{\tfrac{1}{2}}{\tfrac{1}{2}ix^{2}} = e^{\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{1}{4}+\tfrac{1}{2}ia}{\tfrac{1}{2}}{-\tfrac{1}{2}ix^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{-\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{1}{4}-\tfrac{1}{2}ia}{\tfrac{1}{2}}{\tfrac{1}{2}ix^{2}} = e^{\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{1}{4}+\tfrac{1}{2}ia}{\tfrac{1}{2}}{-\tfrac{1}{2}ix^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>exp(-(1)/(4)*I*(x)^(2))*KummerM((1)/(4)-(1)/(2)*I*a, (1)/(2), (1)/(2)*I*(x)^(2)) = exp((1)/(4)*I*(x)^(2))*KummerM((1)/(4)+(1)/(2)*I*a, (1)/(2), -(1)/(2)*I*(x)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[-Divide[1,4]*I*(x)^(2)]*Hypergeometric1F1[Divide[1,4]-Divide[1,2]*I*a, Divide[1,2], Divide[1,2]*I*(x)^(2)] == Exp[Divide[1,4]*I*(x)^(2)]*Hypergeometric1F1[Divide[1,4]+Divide[1,2]*I*a, Divide[1,2], -Divide[1,2]*I*(x)^(2)]</syntaxhighlight> || Failure || Successful || Successful [Tested: 18] || Successful [Tested: 18]
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| | [https://dlmf.nist.gov/12.14.E16 12.14.E16] || [[Item:Q4256|<math>w_{2}(a,x) = xe^{-\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{3}{4}-\tfrac{1}{2}ia}{\tfrac{3}{2}}{\tfrac{1}{2}ix^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>w_{2}(a,x) = xe^{-\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{3}{4}-\tfrac{1}{2}ia}{\tfrac{3}{2}}{\tfrac{1}{2}ix^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(sum(beta[n](a)*((x)^(2*n + 1))/(factorial(2*n + 1)), n = 0..infinity)) = x*exp(-(1)/(4)*I*(x)^(2))*KummerM((3)/(4)-(1)/(2)*I*a, (3)/(2), (1)/(2)*I*(x)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Sum[Subscript[\[Beta], n][a]*Divide[(x)^(2*n + 1),(2*n + 1)!], {n, 0, Infinity}, GenerateConditions->None]) == x*Exp[-Divide[1,4]*I*(x)^(2)]*Hypergeometric1F1[Divide[3,4]-Divide[1,2]*I*a, Divide[3,2], Divide[1,2]*I*(x)^(2)]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/12.14.E16 12.14.E16] || [[Item:Q4256|<math>xe^{-\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{3}{4}-\tfrac{1}{2}ia}{\tfrac{3}{2}}{\tfrac{1}{2}ix^{2}} = xe^{\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{3}{4}+\tfrac{1}{2}ia}{\tfrac{3}{2}}{-\tfrac{1}{2}ix^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>xe^{-\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{3}{4}-\tfrac{1}{2}ia}{\tfrac{3}{2}}{\tfrac{1}{2}ix^{2}} = xe^{\frac{1}{4}ix^{2}}\KummerconfhyperM@{\tfrac{3}{4}+\tfrac{1}{2}ia}{\tfrac{3}{2}}{-\tfrac{1}{2}ix^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>x*exp(-(1)/(4)*I*(x)^(2))*KummerM((3)/(4)-(1)/(2)*I*a, (3)/(2), (1)/(2)*I*(x)^(2)) = x*exp((1)/(4)*I*(x)^(2))*KummerM((3)/(4)+(1)/(2)*I*a, (3)/(2), -(1)/(2)*I*(x)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>x*Exp[-Divide[1,4]*I*(x)^(2)]*Hypergeometric1F1[Divide[3,4]-Divide[1,2]*I*a, Divide[3,2], Divide[1,2]*I*(x)^(2)] == x*Exp[Divide[1,4]*I*(x)^(2)]*Hypergeometric1F1[Divide[3,4]+Divide[1,2]*I*a, Divide[3,2], -Divide[1,2]*I*(x)^(2)]</syntaxhighlight> || Failure || Successful || Successful [Tested: 18] || Successful [Tested: 18]
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| | [https://dlmf.nist.gov/12.14.E17 12.14.E17] || [[Item:Q4257|<math>\paraW@{a}{x} = \sqrt{\frac{2k}{x}}\left(s_{1}(a,x)\cos@@{\omega}-s_{2}(a,x)\sin@@{\omega}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraW@{a}{x} = \sqrt{\frac{2k}{x}}\left(s_{1}(a,x)\cos@@{\omega}-s_{2}(a,x)\sin@@{\omega}\right)</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}+\iunit a)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[(Sqrt[1+Exp[2*Pi*(a)]]-Exp[Pi*(a)])/2] * Exp[Divide[Pi*(a),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 - I*(a), x * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 + I*(a), x * Exp[Divide[Pi*I,4]]] ) == Sqrt[Divide[2*(Sqrt[1 + Exp[2*Pi*a]]- Exp[Pi*a]),x]]*(Subscript[s, 1][a , x]* Cos[Divide[1,4]*(x)^(2)- a*Log[x]+Divide[1,4]*Pi +Divide[1,2]*(Arg[Gamma[Divide[1,2]+ I*a]])]- Subscript[s, 2][a , x]* Sin[Divide[1,4]*(x)^(2)- a*Log[x]+Divide[1,4]*Pi +Divide[1,2]*(Arg[Gamma[Divide[1,2]+ I*a]])])</syntaxhighlight> || Missing Macro Error || Failure || - || Error
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| | [https://dlmf.nist.gov/12.14.E18 12.14.E18] || [[Item:Q4258|<math>\paraW@{a}{-x} = \sqrt{\frac{2}{kx}}\left(s_{1}(a,x)\sin@@{\omega}+s_{2}(a,x)\cos@@{\omega}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\paraW@{a}{-x} = \sqrt{\frac{2}{kx}}\left(s_{1}(a,x)\sin@@{\omega}+s_{2}(a,x)\cos@@{\omega}\right)</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}+\iunit a)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[(Sqrt[1+Exp[2*Pi*(a)]]-Exp[Pi*(a)])/2] * Exp[Divide[Pi*(a),4]] * ( Exp[I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 - I*(a), - x * Exp[-Divide[Pi*I,4]]] + Exp[-I*(Pi/8 + Arg[GAMMA[1/2 + I*(a)]]/2)] * ParabolicCylinderD[- 1/2 + I*(a), - x * Exp[Divide[Pi*I,4]]] ) == Sqrt[Divide[2,k*x]]*(Subscript[s, 1][a , x]* Sin[Divide[1,4]*(x)^(2)- a*Log[x]+Divide[1,4]*Pi +Divide[1,2]*(Arg[Gamma[Divide[1,2]+ I*a]])]+ Subscript[s, 2][a , x]* Cos[Divide[1,4]*(x)^(2)- a*Log[x]+Divide[1,4]*Pi +Divide[1,2]*(Arg[Gamma[Divide[1,2]+ I*a]])])</syntaxhighlight> || Missing Macro Error || Failure || - || Error
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| | [https://dlmf.nist.gov/12.14.E24 12.14.E24] || [[Item:Q4264|<math>\deriv[2]{w}{t} = \mu^{4}(1-t^{2})w</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{w}{t} = \mu^{4}(1-t^{2})w</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [t$(2)]) = (mu)^(4)*(1 - (t)^(2))*w</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {t, 2}] == \[Mu]^(4)*(1 - (t)^(2))*w</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.082531755+.6250000011*I
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| Test Values: {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.6250000011-1.082531755*I
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| Test Values: {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.0825317547305482, 0.6250000000000002]
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| Test Values: {Rule[t, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.0825317547305482, 0.6250000000000002]
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| Test Values: {Rule[t, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.15.E1 12.15.E1] || [[Item:Q4281|<math>\deriv[2]{w}{z}+\left(\nu+\lambda^{-1}-\lambda^{-2}z^{\lambda}\right)w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv[2]{w}{z}+\left(\nu+\lambda^{-1}-\lambda^{-2}z^{\lambda}\right)w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(w, [z$(2)])+(nu + (lambda)^(- 1)- (lambda)^(- 2)* (z)^(lambda))*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[w, {z, 2}]+(\[Nu]+ \[Lambda]^(- 1)- \[Lambda]^(- 2)* (z)^\[Lambda])*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .7322275248+.9199723429*I
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| Test Values: {lambda = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.402820433+.5288298490*I
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| Test Values: {lambda = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.7322275239543282, 0.91997234266967]
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| Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.6337978798301105, 0.5539469388852316]
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| Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[λ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| | [https://dlmf.nist.gov/12.17.E4 12.17.E4] || [[Item:Q4287|<math>\frac{1}{\xi^{2}+\eta^{2}}\left(\pderiv[2]{w}{\xi}+\pderiv[2]{w}{\eta}\right)+\pderiv[2]{w}{\zeta}+k^{2}w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\xi^{2}+\eta^{2}}\left(\pderiv[2]{w}{\xi}+\pderiv[2]{w}{\eta}\right)+\pderiv[2]{w}{\zeta}+k^{2}w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(1)/((xi)^(2)+ (eta)^(2))*(diff(w, [xi$(2)])+ diff(w, [eta$(2)]))+ diff(w, [zeta$(2)])+ (k)^(2)* w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,\[Xi]^(2)+ \[Eta]^(2)]*(D[w, {\[Xi], 2}]+ D[w, {\[Eta], 2}])+ D[w, {\[Zeta], 2}]+ (k)^(2)* w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .8660254040+.5000000000*I
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| Test Values: {eta = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.464101616+2.*I
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| Test Values: {eta = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, xi = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I, k = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.8660254037844387, 0.49999999999999994]
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| Test Values: {Rule[k, 1], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[3.464101615137755, 1.9999999999999998]
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| Test Values: {Rule[k, 2], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[η, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| |}
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| </div> | | </div> |