Results of Parabolic Cylinder Functions: Difference between revisions

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Latest revision as of 17:04, 25 May 2021

Notation: 12.1 Special Notation
Properties: 12.2 Differential Equations
12.3 Graphics
12.4 Power-Series Expansions
12.5 Integral Representations
12.6 Continued Fraction
12.7 Relations to Other Functions
12.8 Recurrence Relations and Derivatives
12.9 Asymptotic Expansions for Large Variable
12.10 Uniform Asymptotic Expansions for Large Parameter
12.11 Zeros
12.12 Integrals
12.13 Sums
12.14 The Function Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \paraW@{a}{x}}
12.15 Generalized Parabolic Cylinder Functions
Applications: 12.16 Mathematical Applications
12.17 Physical Applications
Computation: 12.18 Methods of Computation
12.19 Tables
12.20 Approximations
12.21 Software

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+<div style="-moz-column-count:2; column-count:2;">
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+; Notation : [[12.1|12.1 Special Notation]]<br>
-! DLMF !! Formula !! Maple !! Mathematica !! Symbolic<br>Maple !! Symbolic<br>Mathematica !! Numeric<br>Maple !! Numeric<br>Mathematica
+; 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>
-|-
+; Applications : [[12.16|12.16 Mathematical Applications]]<br>[[12.17|12.17 Physical Applications]]<br>
-| [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>]] || <code>diff(w, [z$(2)])-((1)/(4)*(z)^(2)+ a)* w = 0</code> || <code>D[w, {z, 2}]-(Divide[1,4]*(z)^(2)+ a)* w == 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[1.299038106+.4999999999*I <- {a = -3/2, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</code><br><code>1.299038106+1.000000000*I <- {a = -3/2, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[1.2990381056766582, 0.4999999999999999] <- {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[1.299038105676658, 0.9999999999999999] <- {Rule[a, -1.5], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
+; Computation : [[12.18|12.18 Methods of Computation]]<br>[[12.19|12.19 Tables]]<br>[[12.20|12.20 Approximations]]<br>[[12.21|12.21 Software]]<br>
-|-
+</div>
-| [https://dlmf.nist.gov/12.2.E3 12.2.E3] || [[Item:Q4088|<math>\deriv[2]{w}{z}+\left(\tfrac{1}{4}z^{2}-a\right)w = 0</math>]] || <code>diff(w, [z$(2)])+((1)/(4)*(z)^(2)- a)* w = 0</code> || <code>D[w, {z, 2}]+(Divide[1,4]*(z)^(2)- a)* w == 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[1.299038106+1.000000000*I <- {a = -3/2, w = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</code><br><code>1.299038106+.4999999999*I <- {a = -3/2, w = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[1.299038105676658, 0.9999999999999999] <- {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]]]}</code><br><code>Complex[1.2990381056766582, 0.4999999999999999] <- {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]]]}</code><br></div></div>
-|-
-| [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>]] || <code>diff(w, [z$(2)])+(nu +(1)/(2)-(1)/(4)*(z)^(2))* w = 0</code> || <code>D[w, {z, 2}]+(\[Nu]+Divide[1,2]-Divide[1,4]*(z)^(2))* w == 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.9330127024+.8660254039*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}</code><br><code>.9330127024+1.366025404*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)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [296 / 300]<div class="mw-collapsible-content"><code>{Complex[0.9330127018922196, 0.8660254037844386] <- {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]]]}</code><br><code>Complex[-0.4330127018922191, 0.5000000000000001] <- {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]]]}</code><br></div></div>
-|-
-| [https://dlmf.nist.gov/12.2.E5 12.2.E5] || [[Item:Q4090|<math>\WhittakerparaD{\nu}@{z} = \paraU@{-\tfrac{1}{2}-\nu}{z}</math>]] || <code>CylinderD(nu, z) = CylinderU(-(1)/(2)- nu, z)</code> || <code>ParabolicCylinderD[\[Nu], z] == ParabolicCylinderD[- 1/2 -(-Divide[1,2]- \[Nu]), z]</code> || Successful || Successful || - || Successful [Tested: 70]
-|-
-| [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>]] || <code>CylinderU(a, 0) = (sqrt(Pi))/((2)^((1)/(2)*a +(1)/(4))* GAMMA((3)/(4)+(1)/(2)*a))</code> || <code>ParabolicCylinderD[- 1/2 -(a), 0] == Divide[Sqrt[Pi],(2)^(Divide[1,2]*a +Divide[1,4])* Gamma[Divide[3,4]+Divide[1,2]*a]]</code> || Successful || Successful || - || Successful [Tested: 4]
-|-
-| [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>]] || <code>subs( temp=0, diff( CylinderU(a, temp), temp$(1) ) ) = -(sqrt(Pi))/((2)^((1)/(2)*a -(1)/(4))* GAMMA((1)/(4)+(1)/(2)*a))</code> || <code>(D[ParabolicCylinderD[- 1/2 -(a), temp], {temp, 1}]/.temp-> 0) == -Divide[Sqrt[Pi],(2)^(Divide[1,2]*a -Divide[1,4])* Gamma[Divide[1,4]+Divide[1,2]*a]]</code> || Successful || Successful || - || Successful [Tested: 3]
-|-
-| [https://dlmf.nist.gov/12.2.E8 12.2.E8] || [[Item:Q4093|<math>\paraV@{a}{0} = \frac{\pi 2^{\frac{1}{2}a+\frac{1}{4}}}{\left(\EulerGamma@{\frac{3}{4}-\frac{1}{2}a}\right)^{2}\EulerGamma@{\frac{1}{4}+\frac{1}{2}a}}</math>]] || <code>CylinderV(a, 0) = (Pi*(2)^((1)/(2)*a +(1)/(4)))/((GAMMA((3)/(4)-(1)/(2)*a))^(2)* GAMMA((1)/(4)+(1)/(2)*a))</code> || <code>Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, 0] + ParabolicCylinderD[-(a) - 1/2, -(0)]) == Divide[Pi*(2)^(Divide[1,2]*a +Divide[1,4]),(Gamma[Divide[3,4]-Divide[1,2]*a])^(2)* Gamma[Divide[1,4]+Divide[1,2]*a]]</code> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><code>{Plus[-0.7978845608028653, Times[0.7978845608028655, GAMMA[1.0]]] <- {Rule[a, 0.5]}</code><br></div></div>
-|-
-| [https://dlmf.nist.gov/12.2.E9 12.2.E9] || [[Item:Q4094|<math>\paraV'@{a}{0} = \frac{\pi 2^{\frac{1}{2}a+\frac{3}{4}}}{\left(\EulerGamma@{\frac{1}{4}-\frac{1}{2}a}\right)^{2}\EulerGamma@{\frac{3}{4}+\frac{1}{2}a}}</math>]] || <code>subs( temp=0, diff( CylinderV(a, temp), temp$(1) ) ) = (Pi*(2)^((1)/(2)*a +(3)/(4)))/((GAMMA((1)/(4)-(1)/(2)*a))^(2)* GAMMA((3)/(4)+(1)/(2)*a))</code> || <code>(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]]</code> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><code>{-0.7978845608028653 <- {Rule[a, -0.5]}</code><br></div></div>
-|-
-| [https://dlmf.nist.gov/12.2.E10 12.2.E10] || [[Item:Q4095|<math>\Wronskian@{\paraU@{a}{z},\paraV@{a}{z}} = \sqrt{2/\pi}</math>]] || <code>(CylinderU(a, z))*diff(CylinderV(a, z), z)-diff(CylinderU(a, z), z)*(CylinderV(a, z)) = sqrt(2/ Pi)</code> || <code>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]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [14 / 42]<div class="mw-collapsible-content"><code>14/42]: [[.708254234e-1-.722805450e-2*I <- {a = 3/2, z = 1/2*3^(1/2)+1/2*I}</code><br><code>.4257865765+.241883787*I <- {a = 3/2, z = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><code>{Plus[-0.7978845608028654, Times[Complex[-3.533949646070574*^-17, -3.533949646070574*^-17], GAMMA[-1.0]]] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[-0.7978845608028654, Times[Complex[0.0, -2.1203697876423444*^-16], GAMMA[-1.0]]] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
-|-
-| [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>]] || <code>(CylinderU(a, z))*diff(CylinderU(a, - z), z)-diff(CylinderU(a, z), z)*(CylinderU(a, - z)) = (sqrt(2*Pi))/(GAMMA((1)/(2)+ a))</code> || <code>Wronskian[{ParabolicCylinderD[- 1/2 -(a), z], ParabolicCylinderD[- 1/2 -(a), - z]}, z] == Divide[Sqrt[2*Pi],Gamma[Divide[1,2]+ a]]</code> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
-|-
-| [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>]] || <code>(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)))</code> || <code>Wronskian[{ParabolicCylinderD[- 1/2 -(a), z], ParabolicCylinderD[- 1/2 -(- a), + I*z]}, z] == - I*Exp[+ I*Pi*(Divide[1,2]*a +Divide[1,4])]</code> || Failure || Failure || Successful [Tested: 42] || Successful [Tested: 42]
-|-
-| [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>]] || <code>(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)))</code> || <code>Wronskian[{ParabolicCylinderD[- 1/2 -(a), z], ParabolicCylinderD[- 1/2 -(- a), - I*z]}, z] == + I*Exp[- I*Pi*(Divide[1,2]*a +Divide[1,4])]</code> || Failure || Failure || Successful [Tested: 42] || Successful [Tested: 42]
-|-
-| [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>]] || <code>CylinderU(- n -(1)/(2), - z) = (- 1)^(n)* CylinderU(- n -(1)/(2), z)</code> || <code>ParabolicCylinderD[- 1/2 -(- n -Divide[1,2]), - z] == (- 1)^(n)* ParabolicCylinderD[- 1/2 -(- n -Divide[1,2]), z]</code> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
-|-
-| [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>]] || <code>CylinderV(n +(1)/(2), - z) = (- 1)^(n)* CylinderV(n +(1)/(2), z)</code> || <code>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)])</code> || Successful || Failure || - || Successful [Tested: 21]
-|-
-| [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>]] || <code>CylinderU(a, - z) = - sin(Pi*a)*CylinderU(a, z)+(Pi)/(GAMMA((1)/(2)+ a))*CylinderV(a, z)</code> || <code>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)])</code> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><code>{Plus[Complex[2.097331412545913, 1.9154557103012664], Times[Complex[-2.097331412545913, -1.9154557103012664], GAMMA[2.0]]] <- {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[Complex[-0.668689589092481, 2.108602350101492], Times[Complex[0.668689589092481, -2.108602350101492], GAMMA[2.0]]] <- {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
-|-
-| [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>]] || <code>CylinderV(a, - z) = (cos(Pi*a))/(GAMMA((1)/(2)- a))*CylinderU(a, z)+ sin(Pi*a)*CylinderV(a, z)</code> || <code>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)])</code> || Failure || Failure || Successful [Tested: 21] || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 21]<div class="mw-collapsible-content"><code>{Plus[Complex[-0.3494376482945125, -0.44804866867585064], Times[Complex[0.1478618109503913, 0.18958829384201614], GAMMA[-1.5]]] <- {Rule[a, -2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[Complex[1.1936070900897449, -0.06991225535058408], Times[Complex[-0.5050655153080368, 0.029582824673347826], GAMMA[-1.5]]] <- {Rule[a, -2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
-|-
-| [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>]] || <code>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))</code> || <code>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])</code> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
-|-
-| [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>]] || <code>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))</code> || <code>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])</code> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
-|-
-| [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>]] || <code>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))</code> || <code>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])</code> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
-|-
-| [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>]] || <code>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))</code> || <code>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])</code> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
-|-
-| [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>]] || <code>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)</code> || <code>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]</code> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
-|-
-| [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>]] || <code>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)</code> || <code>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]</code> || Failure || Failure || Successful [Tested: 21] || Successful [Tested: 21]
-|-
-| [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>]] || <code>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)</code> || <code>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]</code> || Failure || Failure || Successful [Tested: 21] || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><code>{Plus[Complex[0.4621744673825597, -0.43960813814518984], Times[Complex[3.533949646070574*^-17, 0.0], GAMMA[-1.0]]] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[Complex[1.0415095884926804, 0.5968092652227893], Times[Complex[1.0601848938211722*^-16, 3.533949646070574*^-17], GAMMA[-1.0]]] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
-|-
-| [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>]] || <code>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)</code> || <code>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]</code> || Failure || Failure || Successful [Tested: 21] || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><code>{Plus[Complex[0.4621744673825599, -0.4396081381451897], Times[Complex[3.533949646070574*^-17, 0.0], GAMMA[-1.0]]] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[Complex[1.0415095884926797, 0.5968092652227891], Times[Complex[1.0601848938211722*^-16, 3.533949646070574*^-17], GAMMA[-1.0]]] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
-|-
-| [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>]] || <code>CylinderU(a, z) = CylinderU(a, 0)*u[1]*(a , z)+ subs( temp=0, diff( CylinderU(a, temp), temp$(1) ) )*u[2]*(a , z)</code> || <code>ParabolicCylinderD[- 1/2 -(a), z] == ParabolicCylinderD[- 1/2 -(a), 0]*Subscript[u, 1]*(a , z)+ (D[ParabolicCylinderD[- 1/2 -(a), temp], {temp, 1}]/.temp-> 0)*Subscript[u, 2]*(a , z)</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><code>{Plus[Complex[0.8412106300093095, 0.2667685495532514], Times[Complex[-0.8618940502981999, 0.18957697416081104], Plus[Complex[0.8660254037844387, 0.49999999999999994], …]]] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[Complex[-0.7641562690755331, 0.8367141764786244], Times[Complex[-1.1066938670748312, -0.24342165324123666], Plus[Complex[-0.4999999999999998, 0.8660254037844387], …]]] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
-|-
-| [https://dlmf.nist.gov/12.4.E2 12.4.E2] || [[Item:Q4110|<math>\paraV@{a}{z} = \paraV@{a}{0}u_{1}(a,z)+\paraV'@{a}{0}u_{2}(a,z)</math>]] || <code>CylinderV(a, z) = CylinderV(a, 0)*u[1]*(a , z)+ subs( temp=0, diff( CylinderV(a, temp), temp$(1) ) )*u[2]*(a , z)</code> || <code>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)])*Subscript[u, 1]*(a , z)+ (D[Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, temp] + ParabolicCylinderD[-(a) - 1/2, -(temp)]), {temp, 1}]/.temp-> 0)*Subscript[u, 2]*(a , z)</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [40 / 42]<div class="mw-collapsible-content"><code>{Plus[0.0, Times[Complex[3.533949646070574*^-17, 0.0], GAMMA[-1.0]]] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[0.0, Times[Complex[1.0601848938211722*^-16, 3.533949646070574*^-17], GAMMA[-1.0]]] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
-|-
-| [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>]] || <code>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)</code> || <code>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]</code> || Successful || Successful || - || Successful [Tested: 21]
-|-
-| [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>]] || <code>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)</code> || <code>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]</code> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 15]<div class="mw-collapsible-content"><code>2/15]: [[Float(infinity)+Float(infinity)*I <- {a = 2, z = 1/2*3^(1/2)+1/2*I}</code><br><code>Float(infinity)+Float(infinity)*I <- {a = 2, z = 1/2-1/2*I*3^(1/2)}</code><br></div></div> || Successful [Tested: 15]
-|-
-| [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>]] || <code>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)</code> || <code>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]</code> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 20]<div class="mw-collapsible-content"><code>2/20]: [[Float(infinity)+Float(infinity)*I <- {a = 3/2, z = 1/2*3^(1/2)+1/2*I}</code><br><code>Float(infinity)+Float(infinity)*I <- {a = 3/2, z = 1/2-1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [5 / 20]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[a, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Indeterminate <- {Rule[a, -0.5], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}</code><br></div></div>
-|-
-| [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>]] || <code>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)</code> || <code>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]</code> || Successful || Failure || - || Successful [Tested: 7]
-|-
-| [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>]] || <code>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 +))</code> || <code>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]</code> || Error || Failure || - || Error
-|-
-| [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>]] || <code>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)</code> || <code>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]</code> || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [126 / 126]<div class="mw-collapsible-content"><code>126/126]: [[.8412106295+.2667685493*I <- {a = -3/2, c = 3/2, z = 1/2*3^(1/2)+1/2*I}</code><br><code>-.7641562685+.8367141760*I <- {a = -3/2, c = 3/2, z = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || Skipped - Because timed out
-|-
-| [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>]] || <code>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)</code> || <code>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]</code> || Failure || Error || Skipped - Because timed out || Skipped - Because timed out
-|-
-| [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>]] || <code>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)</code> || <code>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]</code> || Failure || Error || Skipped - Because timed out || Skipped - Because timed out
-|-
-| [https://dlmf.nist.gov/12.7.E1 12.7.E1] || [[Item:Q4124|<math>\paraU@{-\tfrac{1}{2}}{z} = \WhittakerparaD{0}@{z}</math>]] || <code>CylinderU(-(1)/(2), z) = CylinderD(0, z)</code> || <code>ParabolicCylinderD[- 1/2 -(-Divide[1,2]), z] == ParabolicCylinderD[0, z]</code> || Successful || Successful || - || Successful [Tested: 7]
-|-
-| [https://dlmf.nist.gov/12.7.E1 12.7.E1] || [[Item:Q4124|<math>\WhittakerparaD{0}@{z} = e^{-\frac{1}{4}z^{2}}</math>]] || <code>CylinderD(0, z) = exp(-(1)/(4)*(z)^(2))</code> || <code>ParabolicCylinderD[0, z] == Exp[-Divide[1,4]*(z)^(2)]</code> || Successful || Successful || - || Successful [Tested: 7]
-|-
-| [https://dlmf.nist.gov/12.7.E2 12.7.E2] || [[Item:Q4125|<math>\paraU@{-n-\tfrac{1}{2}}{z} = \WhittakerparaD{n}@{z}</math>]] || <code>CylinderU(- n -(1)/(2), z) = CylinderD(n, z)</code> || <code>ParabolicCylinderD[- 1/2 -(- n -Divide[1,2]), z] == ParabolicCylinderD[n, z]</code> || Successful || Successful || - || Successful [Tested: 7]
-|-
-| [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>]] || <code>CylinderV(-(1)/(2), z) = ((2)/(sqrt(Pi)))* exp((1)/(4)*(z)^(2))*dawson(z/(sqrt(2)))</code> || <code>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])]</code> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><code>{Complex[-0.6813729414422256, -0.33849358809725466] <- {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[0.4709386394349885, -0.6804499612300876] <- {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
-|-
-| [https://dlmf.nist.gov/12.7.E5 12.7.E5] || [[Item:Q4128|<math>\paraU@{\tfrac{1}{2}}{z} = \WhittakerparaD{-1}@{z}</math>]] || <code>CylinderU((1)/(2), z) = CylinderD(- 1, z)</code> || <code>ParabolicCylinderD[- 1/2 -(Divide[1,2]), z] == ParabolicCylinderD[- 1, z]</code> || Successful || Successful || - || Successful [Tested: 7]
-|-
-| [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>]] || <code>CylinderD(- 1, z) = sqrt((1)/(2)*Pi)*exp((1)/(4)*(z)^(2))*erfc(z/(sqrt(2)))</code> || <code>ParabolicCylinderD[- 1, z] == Sqrt[Divide[1,2]*Pi]*Exp[Divide[1,4]*(z)^(2)]*Erfc[z/(Sqrt[2])]</code> || Successful || Successful || - || Successful [Tested: 7]
-|-
-| [https://dlmf.nist.gov/12.7.E6 12.7.E6] || [[Item:Q4129|<math>\paraU@{n+\tfrac{1}{2}}{z} = \WhittakerparaD{-n-1}@{z}</math>]] || <code>CylinderU(n +(1)/(2), z) = CylinderD(- n - 1, z)</code> || <code>ParabolicCylinderD[- 1/2 -(n +Divide[1,2]), z] == ParabolicCylinderD[- n - 1, z]</code> || Successful || Successful || - || Successful [Tested: 7]
-|-
-| [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>]] || <code>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)])</code> || <code>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}]</code> || Failure || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><code>{Plus[0.017848622575954935, Times[0.7141168694348256, DifferenceRoot[Function[{, } <- {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]}</code><br><code>Plus[0.1293114227985036, Times[1.1773796724029832, DifferenceRoot[Function[{, } <- {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]}</code><br></div></div>
-|-
-| [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>]] || <code>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)))</code> || <code>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)])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><code>2/7]: [[-2.928712959+.1903824416*I <- {z = -1/2+1/2*I*3^(1/2)}</code><br><code>-1.578570932+.7263102924*I <- {z = -1/2*3^(1/2)-1/2*I}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><code>{Complex[-2.928712959362369, 0.19038244130086163] <- {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br><code>Complex[-1.5785709321816723, 0.7263102922437361] <- {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</code><br></div></div>
-|-
-| [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>]] || <code>CylinderU(- 1, z) = ((z)^(3/ 2))/(2*sqrt(2*Pi))*(BesselK((1)/(4), (1)/(4)*(z)^(2))+ BesselK((3)/(4), (1)/(4)*(z)^(2)))</code> || <code>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)])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><code>2/7]: [[.5254625443+1.913964596*I <- {z = -1/2+1/2*I*3^(1/2)}</code><br><code>-.1061142274-1.367750447*I <- {z = -1/2*3^(1/2)-1/2*I}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><code>{Complex[0.5254625445137794, 1.9139645960722755] <- {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br><code>Complex[-0.10611422720224939, -1.3677504477251] <- {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</code><br></div></div>
-|-
-| [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>]] || <code>CylinderU(0, z) = sqrt((z)/(2*Pi))*BesselK((1)/(4), (1)/(4)*(z)^(2))</code> || <code>ParabolicCylinderD[- 1/2 -(0), z] == Sqrt[Divide[z,2*Pi]]*BesselK[Divide[1,4], Divide[1,4]*(z)^(2)]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><code>2/7]: [[2.016879450-1.384601654*I <- {z = -1/2+1/2*I*3^(1/2)}</code><br><code>1.973186649+1.022506910*I <- {z = -1/2*3^(1/2)-1/2*I}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><code>{Complex[2.0168794499257325, -1.3846016541017099] <- {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br><code>Complex[1.9731866495584476, 1.0225069102497304] <- {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</code><br></div></div>
-|-
-| [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>]] || <code>CylinderU(1, z) = ((z)^(3/ 2))/(sqrt(2*Pi))*(BesselK((3)/(4), (1)/(4)*(z)^(2))- BesselK((1)/(4), (1)/(4)*(z)^(2)))</code> || <code>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)])</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><code>2/7]: [[.6696041257-1.050010143*I <- {z = -1/2+1/2*I*3^(1/2)}</code><br><code>2.182924166+1.008719675*I <- {z = -1/2*3^(1/2)-1/2*I}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><code>{Complex[0.6696041258052213, -1.050010141970097] <- {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br><code>Complex[2.1829241651976083, 1.0087196737510498] <- {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</code><br></div></div>
-|-
-| [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>]] || <code>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))</code> || <code>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)]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 42]<div class="mw-collapsible-content"><code>10/42]: [[-1.528312538+1.673428352*I <- {a = -3/2, z = -1/2+1/2*I*3^(1/2)}</code><br><code>-1.682421259-.5335370987*I <- {a = -3/2, z = -1/2*3^(1/2)-1/2*I}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 42]<div class="mw-collapsible-content"><code>{Complex[-1.5283125381510665, 1.6734283529572487] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br><code>Complex[-1.6824212600186188, -0.5335370991065028] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</code><br></div></div>
-|-
-| [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>]] || <code>(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))</code> || <code>(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)]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 42]<div class="mw-collapsible-content"><code>12/42]: [[1.528312538-1.673428353*I <- {a = -3/2, z = -1/2+1/2*I*3^(1/2)}</code><br><code>1.682421260+.5335370988*I <- {a = -3/2, z = -1/2*3^(1/2)-1/2*I}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 42]<div class="mw-collapsible-content"><code>{Complex[1.5283125381510665, -1.673428352957249] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br><code>Complex[1.6824212600186188, 0.5335370991065027] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</code><br></div></div>
-|-
-| [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>]] || <code>(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))</code> || <code>(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)]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 42]<div class="mw-collapsible-content"><code>12/42]: [[.725579081e-1+1.600870446*I <- {a = -3/2, z = -1/2+1/2*I*3^(1/2)}</code><br><code>-.5744420805-1.107979180*I <- {a = -3/2, z = -1/2*3^(1/2)-1/2*I}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 42]<div class="mw-collapsible-content"><code>{Complex[0.0725579074030912, 1.600870445554158] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br><code>Complex[-0.574442080456058, -1.1079791795625606] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</code><br></div></div>
-|-
-| [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>]] || <code>(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))</code> || <code>(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)]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 42]<div class="mw-collapsible-content"><code>12/42]: [[.725579081e-1+1.600870446*I <- {a = -3/2, z = -1/2+1/2*I*3^(1/2)}</code><br><code>-.5744420804-1.107979180*I <- {a = -3/2, z = -1/2*3^(1/2)-1/2*I}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 42]<div class="mw-collapsible-content"><code>{Complex[0.0725579074030912, 1.600870445554158] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br><code>Complex[-0.5744420804560579, -1.1079791795625609] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</code><br></div></div>
-|-
-| [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>]] || <code>z*CylinderU(a, z)- CylinderU(a - 1, z)+(a +(1)/(2))* CylinderU(a + 1, z) = 0</code> || <code>z*ParabolicCylinderD[- 1/2 -(a), z]- ParabolicCylinderD[- 1/2 -(a - 1), z]+(a +Divide[1,2])* ParabolicCylinderD[- 1/2 -(a + 1), z] == 0</code> || Successful || Successful || - || Successful [Tested: 42]
-|-
-| [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>]] || <code>diff( CylinderU(a, z), z$(1) )+(1)/(2)*z*CylinderU(a, z)+(a +(1)/(2))* CylinderU(a + 1, z) = 0</code> || <code>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</code> || Successful || Successful || - || Successful [Tested: 42]
-|-
-| [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>]] || <code>diff( CylinderU(a, z), z$(1) )-(1)/(2)*z*CylinderU(a, z)+ CylinderU(a - 1, z) = 0</code> || <code>D[ParabolicCylinderD[- 1/2 -(a), z], {z, 1}]-Divide[1,2]*z*ParabolicCylinderD[- 1/2 -(a), z]+ ParabolicCylinderD[- 1/2 -(a - 1), z] == 0</code> || Successful || Successful || - || Successful [Tested: 42]
-|-
-| [https://dlmf.nist.gov/12.8.E4 12.8.E4] || [[Item:Q4141|<math>2\paraU'@{a}{z}+\paraU@{a-1}{z}+(a+\tfrac{1}{2})\paraU@{a+1}{z} = 0</math>]] || <code>2*diff( CylinderU(a, z), z$(1) )+ CylinderU(a - 1, z)+(a +(1)/(2))* CylinderU(a + 1, z) = 0</code> || <code>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</code> || Successful || Successful || - || Successful [Tested: 42]
-|-
-| [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>]] || <code>z*CylinderV(a, z)- CylinderV(a + 1, z)+(a -(1)/(2))* CylinderV(a - 1, z) = 0</code> || <code>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</code> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><code>{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]]] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>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]]] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
-|-
-| [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>]] || <code>diff( CylinderV(a, z), z$(1) )-(1)/(2)*z*CylinderV(a, z)-(a -(1)/(2))* CylinderV(a - 1, z) = 0</code> || <code>D[Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]), {z, 1}]-Divide[1,2]*z*Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)])-(a -Divide[1,2])* Divide[GAMMA[1/2 + a - 1], Pi]*(Sin[Pi*(a - 1)] * ParabolicCylinderD[-(a - 1) - 1/2, z] + ParabolicCylinderD[-(a - 1) - 1/2, -(z)]) == 0</code> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [39 / 42]<div class="mw-collapsible-content"><code>{Plus[Complex[0.0, 0.0], Times[Complex[-7.067899292141149*^-17, 0.0], GAMMA[-2.0]], Times[Complex[-1.5302450845960716*^-17, -8.834874115176434*^-18], GAMMA[-1.0]]] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[Times[Complex[-1.4135798584282297*^-16, 0.0], GAMMA[-2.0]], Times[Complex[-9.955091265133296*^-17, -1.7329819619999673*^-18], GAMMA[-1.0]]] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
-|-
-| [https://dlmf.nist.gov/12.8.E7 12.8.E7] || [[Item:Q4144|<math>\paraV'@{a}{z}+\tfrac{1}{2}z\paraV@{a}{z}-\paraV@{a+1}{z} = 0</math>]] || <code>diff( CylinderV(a, z), z$(1) )+(1)/(2)*z*CylinderV(a, z)- CylinderV(a + 1, z) = 0</code> || <code>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</code> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><code>{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]]] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[Times[Complex[-1.83165059034313*^-16, 7.241197488341145*^-17], GAMMA[-1.0]], Times[Complex[-7.067899292141149*^-17, -8.834874115176436*^-18], GAMMA[0.0]]] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
-|-
-| [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>]] || <code>2*diff( CylinderV(a, z), z$(1) )- CylinderV(a + 1, z)-(a -(1)/(2))* CylinderV(a - 1, z) = 0</code> || <code>2*D[Divide[GAMMA[1/2 + a], Pi]*(Sin[Pi*(a)] * ParabolicCylinderD[-(a) - 1/2, z] + ParabolicCylinderD[-(a) - 1/2, -(z)]), {z, 1}]- Divide[GAMMA[1/2 + a + 1], Pi]*(Sin[Pi*(a + 1)] * ParabolicCylinderD[-(a + 1) - 1/2, z] + ParabolicCylinderD[-(a + 1) - 1/2, -(z)])-(a -Divide[1,2])* Divide[GAMMA[1/2 + a - 1], Pi]*(Sin[Pi*(a - 1)] * ParabolicCylinderD[-(a - 1) - 1/2, z] + ParabolicCylinderD[-(a - 1) - 1/2, -(z)]) == 0</code> || Successful || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 42]<div class="mw-collapsible-content"><code>{Plus[Complex[0.0, 0.0], Times[Complex[-7.067899292141149*^-17, 0.0], GAMMA[-2.0]], Times[Complex[0.0, -8.834874115176436*^-18], GAMMA[0.0]]] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[Times[Complex[-1.4135798584282297*^-16, 0.0], GAMMA[-2.0]], Times[Complex[-2.8271597168564594*^-16, 7.067899292141149*^-17], GAMMA[-1.0]], Times[Complex[-7.067899292141149*^-17, -8.834874115176436*^-18], GAMMA[0.0]]] <- {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
-|-
-| [https://dlmf.nist.gov/12.8.E9 12.8.E9] || [[Item:Q4146|<math>\deriv[m]{}{z}\left(e^{\frac{1}{4}z^{2}}\paraU@{a}{z}\right) = (-1)^{m}\Pochhammersym{\tfrac{1}{2}+a}{m}e^{\frac{1}{4}z^{2}}\paraU@{a+m}{z}</math>]] || <code>diff(exp((1)/(4)*(z)^(2))*CylinderU(a, z), [z$(m)]) = (- 1)^(m)* pochhammer((1)/(2)+ a, m)*exp((1)/(4)*(z)^(2))*CylinderU(a + m, z)</code> || <code>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]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [96 / 126]<div class="mw-collapsible-content"><code>{Plus[Complex[-1.0, 0.0], 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]]]]]]]}]][1.0]], {Rule[a, -1.5], Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[Complex[0.0, 0.0], Times[2.0, 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]]]}</code><br></div></div>
-|-
-| [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>]] || <code>diff(exp(-(1)/(4)*(z)^(2))*CylinderU(a, z), [z$(m)]) = (- 1)^(m)* exp(-(1)/(4)*(z)^(2))*CylinderU(a - m, z)</code> || <code>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]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [96 / 126]<div class="mw-collapsible-content"><code>{Plus[Complex[-0.07045205979755337, 0.7756076114781977], 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]]]]]}]][1.0]], {Rule[a, -1.5], Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[Complex[2.000032302229117, -0.49556574541480647], Times[2.0, 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]]]}</code><br></div></div>
-|-
-| [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>]] || <code>diff(exp((1)/(4)*(z)^(2))*CylinderV(a, z), [z$(m)]) = exp((1)/(4)*(z)^(2))*CylinderV(a + m, z)</code> || <code>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)])</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [126 / 126]<div class="mw-collapsible-content"><code>{Plus[Times[Complex[2.150599663294456*^-18, -9.777500999643939*^-18], GAMMA[0.0]], Times[0.3183098861837907, GAMMA[-1.0], Plus[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]], 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]]]]]]]}]][1.0], 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]]]]]]]}]][1.0]]]]], {Rule[a, -1.5], Rule[m, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[Times[Complex[-0.9299481905237211, -0.4298894311242862], GAMMA[1.0]], Times[0.6366197723675814, GAMMA[-1.0], Plus[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]], 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]]]}</code><br></div></div>
-|-
-| [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>]] || <code>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)</code> || <code>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)])</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [126 / 126]<div class="mw-collapsible-content"><code>{Plus[Times[Complex[-6.091780348003315*^-17, 1.3399109614774574*^-17], GAMMA[-2.0]], Times[0.3183098861837907, GAMMA[-1.0], Plus[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]], 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]], []], 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]]]}</code><br><code>Plus[Times[Complex[1.6052302675286988*^-15, 3.2948039393826443*^-16], GAMMA[-3.0]], Times[0.6366197723675814, GAMMA[-1.0], Plus[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]], 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]]]}</code><br></div></div>
-|-
-| [https://dlmf.nist.gov/12.10#Ex1 12.10#Ex1] || [[Item:Q4154|<math>a = +\tfrac{1}{2}\mu^{2}</math>]] || <code>a = +(1)/(2)*(mu)^(2)</code> || <code>a == +Divide[1,2]*\[Mu]^(2)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/12.10#Ex2 12.10#Ex2] || [[Item:Q4155|<math>x = \mu t\sqrt{2}</math>]] || <code>x = mu*t*sqrt(2)</code> || <code>x == \[Mu]*t*Sqrt[2]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/12.10.E2 12.10.E2] || [[Item:Q4156|<math>\deriv[2]{w}{t} = \mu^{4}(t^{2}+ 1)w</math>]] || <code>diff(w, [t$(2)]) = (mu)^(4)*((t)^(2)+ 1)* w</code> || <code>D[w, {t, 2}] == \[Mu]^(4)*((t)^(2)+ 1)* w</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[2.814582564-1.625000003*I <- {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = 1/2*3^(1/2)+1/2*I}</code><br><code>1.625000003+2.814582564*I <- {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[2.814582562299425, -1.6250000000000009] <- {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]]]}</code><br><code>Complex[2.814582562299425, -1.6250000000000009] <- {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]]]}</code><br></div></div>
-|-
-| [https://dlmf.nist.gov/12.10.E2 12.10.E2] || [[Item:Q4156|<math>\deriv[2]{w}{t} = \mu^{4}(t^{2}- 1)w</math>]] || <code>diff(w, [t$(2)]) = (mu)^(4)*((t)^(2)- 1)* w</code> || <code>D[w, {t, 2}] == \[Mu]^(4)*((t)^(2)- 1)* w</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[1.082531755-.6250000011*I <- {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = 1/2*3^(1/2)+1/2*I}</code><br><code>.6250000011+1.082531755*I <- {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[1.0825317547305482, -0.6250000000000002] <- {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]]]}</code><br><code>Complex[1.0825317547305482, -0.6250000000000002] <- {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]]]}</code><br></div></div>
-|-
-| [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>]] || <code>v[s]*(t) = u[s]*(t)+(1)/(2)*t*u[s - 1]*(t)- r[s - 2]*(t)</code> || <code>Subscript[v, s]*(t) == Subscript[u, s]*(t)+Divide[1,2]*t*Subscript[u, s - 1]*(t)- Subscript[r, s - 2]*(t)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/12.10#Ex9 12.10#Ex9] || [[Item:Q4175|<math>\gamma_{0} = 1</math>]] || <code>gamma[0] = 1</code> || <code>Subscript[\[Gamma], 0] == 1</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/12.10#Ex10 12.10#Ex10] || [[Item:Q4176|<math>\gamma_{1} = -\tfrac{1}{24}</math>]] || <code>gamma[1] = -(1)/(24)</code> || <code>Subscript[\[Gamma], 1] == -Divide[1,24]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/12.10#Ex11 12.10#Ex11] || [[Item:Q4177|<math>\gamma_{2} = \tfrac{1}{1152}</math>]] || <code>gamma[2] = (1)/(1152)</code> || <code>Subscript[\[Gamma], 2] == Divide[1,1152]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/12.10#Ex12 12.10#Ex12] || [[Item:Q4178|<math>\gamma_{3} = \tfrac{1003}{4\;14720}</math>]] || <code>gamma[3] = (1003)/(414720)</code> || <code>Subscript[\[Gamma], 3] == Divide[1003,414720]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/12.10#Ex13 12.10#Ex13] || [[Item:Q4179|<math>\gamma_{4} = -\tfrac{4027}{398\;13120}</math>]] || <code>gamma[4] = -(4027)/(39813120)</code> || <code>Subscript[\[Gamma], 4] == -Divide[4027,39813120]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [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>]] || <code>A[1]*(tau) = -(1)/(12)*tau*(20*(tau)^(2)+ 30*tau + 9)</code> || <code>Subscript[A, 1]*(\[Tau]) == -Divide[1,12]*\[Tau]*(20*\[Tau]^(2)+ 30*\[Tau]+ 9)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [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>]] || <code>A[2]*(tau) = (1)/(288)*(tau)^(2)*(6160*(tau)^(4)+ 18480*(tau)^(3)+ 19404*(tau)^(2)+ 8028*tau + 945)</code> || <code>Subscript[A, 2]*(\[Tau]) == Divide[1,288]*\[Tau]^(2)*(6160*\[Tau]^(4)+ 18480*\[Tau]^(3)+ 19404*\[Tau]^(2)+ 8028*\[Tau]+ 945)</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [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>]] || <code>A[s]*(zeta) = (zeta)^(- 3*s)* sum(beta[m]*(phi*(zeta))^(6*(2*s - m))* u[2*s - m]*(t), m = 0..2*s)</code> || <code>Subscript[A, s]*(\[Zeta]) == \[Zeta]^(- 3*s)* Sum[Subscript[\[Beta], m]*(\[Phi]*(\[Zeta]))^(6*(2*s - m))* Subscript[u, 2*s - m]*(t), {m, 0, 2*s}, GenerateConditions->None]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [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>]] || <code>(zeta)^(2)* B[s]*(zeta) = - (zeta)^(- 3*s)* sum(alpha[m]*(phi*(zeta))^(6*(2*s - m + 1))* u[2*s - m + 1]*(t), m = 0..2*s + 1)</code> || <code>\[Zeta]^(2)* Subscript[B, s]*(\[Zeta]) == - \[Zeta]^(- 3*s)* Sum[Subscript[\[Alpha], m]*(\[Phi]*(\[Zeta]))^(6*(2*s - m + 1))* Subscript[u, 2*s - m + 1]*(t), {m, 0, 2*s + 1}, GenerateConditions->None]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [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>]] || <code>zeta*C[s]*(zeta) = - (zeta)^(- 3*s)* sum(beta[m]*(phi*(zeta))^(6*(2*s - m + 1))* v[2*s - m + 1]*(t), m = 0..2*s + 1)</code> || <code>\[Zeta]*Subscript[C, s]*(\[Zeta]) == - \[Zeta]^(- 3*s)* Sum[Subscript[\[Beta], m]*(\[Phi]*(\[Zeta]))^(6*(2*s - m + 1))* Subscript[v, 2*s - m + 1]*(t), {m, 0, 2*s + 1}, GenerateConditions->None]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [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>]] || <code>D[s]*(zeta) = (zeta)^(- 3*s)* sum(alpha[m]*(phi*(zeta))^(6*(2*s - m))* v[2*s - m]*(t), m = 0..2*s)</code> || <code>Subscript[D, s]*(\[Zeta]) == \[Zeta]^(- 3*s)* Sum[Subscript[\[Alpha], m]*(\[Phi]*(\[Zeta]))^(6*(2*s - m))* Subscript[v, 2*s - m]*(t), {m, 0, 2*s}, GenerateConditions->None]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/12.11.E5 12.11.E5] || [[Item:Q4221|<math>p_{0}(\zeta) = t(\zeta)</math>]] || <code>p[0]*(zeta) = t*(zeta)</code> || <code>Subscript[p, 0]*(\[Zeta]) == t*(\[Zeta])</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [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>]] || <code>p[1]*(zeta) = ((t)^(3)- 6*t)/(24*((t)^(2)- 1)^(2))+(5)/(48*(((t)^(2)- 1)*(zeta)^(3))^((1)/(2)))</code> || <code>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])]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/12.11.E8 12.11.E8] || [[Item:Q4224|<math>q_{0}(\zeta) = t(\zeta)</math>]] || <code>q[0]*(zeta) = t*(zeta)</code> || <code>Subscript[q, 0]*(\[Zeta]) == t*(\[Zeta])</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [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>]] || <code>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))))</code> || <code>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])]]</code> || Successful || Error || - || Skipped - Because timed out
-|-
-| [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>]] || <code>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)</code> || <code>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]</code> || Failure || Failure || Skipped - Because timed out || Successful [Tested: 2]
-|-
-| [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>]] || <code>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)</code> || <code>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]</code> || Failure || Error || Skipped - Because timed out || Skipped - Because timed out
-|-
-| [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>]] || <code>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)</code> || <code>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]</code> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
-|-
-| [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>]] || <code>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)</code> || <code>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]</code> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
-|-
-| [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>]] || <code>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)</code> || <code>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]</code> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
-|-
-| [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>]] || <code>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)</code> || <code>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]</code> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
-|-
-| [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>]] || <code>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)</code> || <code>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]</code> || Failure || Failure || Skipped - Because timed out || Skip - No test values generated
-|-
-| [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>]] || <code>Error</code> || <code>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])</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 6]<div class="mw-collapsible-content"><code>{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]]]]]]]]]]] <- {Rule[a, -1.5]}</code><br><code>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]]]]]]]]]]] <- {Rule[a, 1.5]}</code><br></div></div>
-|-
-| [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>]] || <code>Error</code> || <code>(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])</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 6]<div class="mw-collapsible-content"><code>{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]]]]]]]]]]]] <- {Rule[a, -1.5]}</code><br><code>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]]]]]]]]]]]] <- {Rule[a, 1.5]}</code><br></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>]] || <code>Error</code> || <code>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[k/ 2]*Exp[Divide[1,4]*Pi*a]*(Exp[I*\[Rho]]*ParabolicCylinderD[- 1/2 -(I*a), x*Exp[- Pi*I/ 4]]+ Exp[- I*\[Rho]]*ParabolicCylinderD[- 1/2 -(- I*a), x*Exp[Pi*I/ 4]])</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><code>{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]]]]]]]]]]] <- {Rule[a, -1.5], Rule[x, 1.5]}</code><br><code>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]]]]]]]]]]] <- {Rule[a, -1.5], Rule[x, 0.5]}</code><br></div></div>
-|-
-| [https://dlmf.nist.gov/12.14#Ex5 12.14#Ex5] || [[Item:Q4249|<math>\alpha_{0}(a) = 1</math>]] || <code>alpha[0]*(a) = 1</code> || <code>Subscript[\[Alpha], 0]*(a) == 1</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/12.14#Ex6 12.14#Ex6] || [[Item:Q4250|<math>\alpha_{1}(a) = a</math>]] || <code>alpha[1]*(a) = a</code> || <code>Subscript[\[Alpha], 1]*(a) == a</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/12.14#Ex7 12.14#Ex7] || [[Item:Q4251|<math>\beta_{0}(a) = 1</math>]] || <code>beta[0]*(a) = 1</code> || <code>Subscript[\[Beta], 0]*(a) == 1</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [https://dlmf.nist.gov/12.14#Ex8 12.14#Ex8] || [[Item:Q4252|<math>\beta_{1}(a) = a</math>]] || <code>beta[1]*(a) = a</code> || <code>Subscript[\[Beta], 1]*(a) == a</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
-|-
-| [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>]] || <code>Error</code> || <code>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)])</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>{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]]]]]]]]]] <- {Rule[x, 1.5]}</code><br><code>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]]]]]]]]]] <- {Rule[x, 0.5]}</code><br></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>]] || <code>Error</code> || <code>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)])</code> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>{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]]]]]]]]]] <- {Rule[x, 1.5]}</code><br><code>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]]]]]]]]]] <- {Rule[x, 0.5]}</code><br></div></div>
-|-
-| [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>]] || <code>w[1]*(a , x) = exp(-(1)/(4)*I*(x)^(2))*KummerM((1)/(4)-(1)/(2)*I*a, (1)/(2), (1)/(2)*I*(x)^(2))</code> || <code>Subscript[w, 1]*(a , x) == 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)]</code> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
-|-
-| [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>]] || <code>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))</code> || <code>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)]</code> || 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>]] || <code>w[2]*(a , x) = x*exp(-(1)/(4)*I*(x)^(2))*KummerM((3)/(4)-(1)/(2)*I*a, (3)/(2), (1)/(2)*I*(x)^(2))</code> || <code>Subscript[w, 2]*(a , x) == 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)]</code> || 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>]] || <code>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))</code> || <code>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)]</code> || 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>]] || <code>Error</code> || <code>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)*Cos[\[Omega]]- Subscript[s, 2]*(a , x)*Sin[\[Omega]])</code> || 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>]] || <code>Error</code> || <code>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[\[Omega]]+ Subscript[s, 2]*(a , x)*Cos[\[Omega]])</code> || 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>]] || <code>diff(w, [t$(2)]) = (mu)^(4)*(1 - (t)^(2))* w</code> || <code>D[w, {t, 2}] == \[Mu]^(4)*(1 - (t)^(2))* w</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-1.082531755+.6250000011*I <- {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = 1/2*3^(1/2)+1/2*I}</code><br><code>-.6250000011-1.082531755*I <- {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, w = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[-1.0825317547305482, 0.6250000000000002] <- {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]]]}</code><br><code>Complex[-1.0825317547305482, 0.6250000000000002] <- {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]]]}</code><br></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>]] || <code>diff(w, [z$(2)])+(nu + (lambda)^(- 1)- (lambda)^(- 2)* (z)^(lambda))* w = 0</code> || <code>D[w, {z, 2}]+(\[Nu]+ \[Lambda]^(- 1)- \[Lambda]^(- 2)* (z)^\[Lambda])* w == 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.7322275248+.9199723429*I <- {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}</code><br><code>1.402820433+.5288298490*I <- {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)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[0.7322275239543282, 0.91997234266967] <- {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]]]}</code><br><code>Complex[-0.6337978798301105, 0.5539469388852316] <- {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]]]}</code><br></div></div>
-|-
-| [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>]] || <code>(1)/((xi)^(2)+ (eta)^(2))*(diff(w, [xi$(2)])+ diff(w, [eta$(2)]))+ diff(w, [zeta$(2)])+ (k)^(2)* w = 0</code> || <code>Divide[1,\[Xi]^(2)+ \[Eta]^(2)]*(D[w, {\[Xi], 2}]+ D[w, {\[Eta], 2}])+ D[w, {\[Zeta], 2}]+ (k)^(2)* w == 0</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[.8660254040+.5000000000*I <- {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}</code><br><code>3.464101616+2.*I <- {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}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[0.8660254037844387, 0.49999999999999994] <- {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]]]}</code><br><code>Complex[3.464101615137755, 1.9999999999999998] <- {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]]]}</code><br></div></div>
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Results of Parabolic Cylinder Functions: Difference between revisions

Latest revision as of 17:04, 25 May 2021

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