32.15: Difference between revisions

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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/32.15.E1 32.15.E1] || [[Item:Q9493|<math>\int_{-\infty}^{\infty}\exp@{-\tfrac{1}{4}\xi^{4}-z\xi^{2}}p_{m}(\xi)p_{n}(\xi)\diff{\xi} = \Kroneckerdelta{m}{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-\infty}^{\infty}\exp@{-\tfrac{1}{4}\xi^{4}-z\xi^{2}}p_{m}(\xi)p_{n}(\xi)\diff{\xi} = \Kroneckerdelta{m}{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(exp(-(1)/(4)*(xi)^(4)- z*(xi)^(2))*p[m](xi)* p[n](xi), xi = - infinity..infinity) = KroneckerDelta[m, n]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[-Divide[1,4]*\[Xi]^(4)- z*\[Xi]^(2)]*Subscript[p, m][\[Xi]]* Subscript[p, n][\[Xi]], {\[Xi], - Infinity, Infinity}, GenerateConditions->None] == KroneckerDelta[m, n]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.5089051774+.3195154069*I
| [https://dlmf.nist.gov/32.15.E1 32.15.E1] || <math qid="Q9493">\int_{-\infty}^{\infty}\exp@{-\tfrac{1}{4}\xi^{4}-z\xi^{2}}p_{m}(\xi)p_{n}(\xi)\diff{\xi} = \Kroneckerdelta{m}{n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{-\infty}^{\infty}\exp@{-\tfrac{1}{4}\xi^{4}-z\xi^{2}}p_{m}(\xi)p_{n}(\xi)\diff{\xi} = \Kroneckerdelta{m}{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>int(exp(-(1)/(4)*(xi)^(4)- z*(xi)^(2))*p[m](xi)* p[n](xi), xi = - infinity..infinity) = KroneckerDelta[m, n]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[-Divide[1,4]*\[Xi]^(4)- z*\[Xi]^(2)]*Subscript[p, m][\[Xi]]* Subscript[p, n][\[Xi]], {\[Xi], - Infinity, Infinity}, GenerateConditions->None] == KroneckerDelta[m, n]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.5089051774+.3195154069*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, p[m] = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, m = 1, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .4910948226+.3195154069*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, p[m] = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, m = 1, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .4910948226+.3195154069*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, p[m] = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, m = 1, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.5089051767265081, 0.31951540648426185]
Test Values: {z = 1/2*3^(1/2)+1/2*I, p[m] = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, m = 1, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.5089051767265081, 0.31951540648426185]
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Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, m], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, m], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/32.15.E2 32.15.E2] || [[Item:Q9494|<math>a_{n+1}(z)p_{n+1}(\xi) = \xi p_{n}(\xi)-a_{n}(z)p_{n-1}(\xi)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a_{n+1}(z)p_{n+1}(\xi) = \xi p_{n}(\xi)-a_{n}(z)p_{n-1}(\xi)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[n + 1](z)* p[n + 1](xi) = xi*p[n](xi)- a[n](z)* p[n - 1](xi)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, n + 1][z]* Subscript[p, n + 1][\[Xi]] == \[Xi]*Subscript[p, n][\[Xi]]- Subscript[a, n][z]* Subscript[p, n - 1][\[Xi]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/32.15.E2 32.15.E2] || <math qid="Q9494">a_{n+1}(z)p_{n+1}(\xi) = \xi p_{n}(\xi)-a_{n}(z)p_{n-1}(\xi)</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a_{n+1}(z)p_{n+1}(\xi) = \xi p_{n}(\xi)-a_{n}(z)p_{n-1}(\xi)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[n + 1](z)* p[n + 1](xi) = xi*p[n](xi)- a[n](z)* p[n - 1](xi)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, n + 1][z]* Subscript[p, n + 1][\[Xi]] == \[Xi]*Subscript[p, n][\[Xi]]- Subscript[a, n][z]* Subscript[p, n - 1][\[Xi]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/32.15.E3 32.15.E3] || [[Item:Q9495|<math>(u_{n+1}+u_{n}+u_{n-1})u_{n} = n-2zu_{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>(u_{n+1}+u_{n}+u_{n-1})u_{n} = n-2zu_{n}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(u[n + 1]+ u[n]+ u[n - 1])*u[n] = n - 2*z*u[n]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(Subscript[u, n + 1]+ Subscript[u, n]+ Subscript[u, n - 1])*Subscript[u, n] == n - 2*z*Subscript[u, n]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/32.15.E3 32.15.E3] || <math qid="Q9495">(u_{n+1}+u_{n}+u_{n-1})u_{n} = n-2zu_{n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>(u_{n+1}+u_{n}+u_{n-1})u_{n} = n-2zu_{n}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(u[n + 1]+ u[n]+ u[n - 1])*u[n] = n - 2*z*u[n]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(Subscript[u, n + 1]+ Subscript[u, n]+ Subscript[u, n - 1])*Subscript[u, n] == n - 2*z*Subscript[u, n]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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Latest revision as of 12:13, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
32.15.E1 ∫ - ∞ ∞ exp ⁡ ( - 1 4 ⁢ ξ 4 - z ⁢ ξ 2 ) ⁢ p m ⁢ ( ξ ) ⁢ p n ⁢ ( ξ ) ⁢ d ξ = δ m , n superscript subscript 1 4 superscript 𝜉 4 𝑧 superscript 𝜉 2 subscript 𝑝 𝑚 𝜉 subscript 𝑝 𝑛 𝜉 𝜉 Kronecker 𝑚 𝑛 {\displaystyle{\displaystyle\int_{-\infty}^{\infty}\exp\left(-\tfrac{1}{4}\xi^% {4}-z\xi^{2}\right)p_{m}(\xi)p_{n}(\xi)\mathrm{d}\xi=\delta_{m,n}}}
\int_{-\infty}^{\infty}\exp@{-\tfrac{1}{4}\xi^{4}-z\xi^{2}}p_{m}(\xi)p_{n}(\xi)\diff{\xi} = \Kroneckerdelta{m}{n}		 

int(exp(-(1)/(4)*(xi)^(4)- z*(xi)^(2))*p[m](xi)* p[n](xi), xi = - infinity..infinity) = KroneckerDelta[m, n]

Integrate[Exp[-Divide[1,4]*\[Xi]^(4)- z*\[Xi]^(2)]*Subscript[p, m][\[Xi]]* Subscript[p, n][\[Xi]], {\[Xi], - Infinity, Infinity}, GenerateConditions->None] == KroneckerDelta[m, n]
Failure Failure
Failed [300 / 300]
Result: -.5089051774+.3195154069*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, p[m] = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, m = 1, n = 1}


Result: .4910948226+.3195154069*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, p[m] = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, m = 1, n = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.5089051767265081, 0.31951540648426185]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, m], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.49109482327349185, 0.31951540648426185]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, m], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
32.15.E2 a n + 1 ⁢ ( z ) ⁢ p n + 1 ⁢ ( ξ ) = ξ ⁢ p n ⁢ ( ξ ) - a n ⁢ ( z ) ⁢ p n - 1 ⁢ ( ξ ) subscript 𝑎 𝑛 1 𝑧 subscript 𝑝 𝑛 1 𝜉 𝜉 subscript 𝑝 𝑛 𝜉 subscript 𝑎 𝑛 𝑧 subscript 𝑝 𝑛 1 𝜉 {\displaystyle{\displaystyle a_{n+1}(z)p_{n+1}(\xi)=\xi p_{n}(\xi)-a_{n}(z)p_{% n-1}(\xi)}}
a_{n+1}(z)p_{n+1}(\xi) = \xi p_{n}(\xi)-a_{n}(z)p_{n-1}(\xi)		 

a[n + 1](z)* p[n + 1](xi) = xi*p[n](xi)- a[n](z)* p[n - 1](xi)
 
Subscript[a, n + 1][z]* Subscript[p, n + 1][\[Xi]] == \[Xi]*Subscript[p, n][\[Xi]]- Subscript[a, n][z]* Subscript[p, n - 1][\[Xi]]
 Skipped - no semantic math Skipped - no semantic math - -
32.15.E3 ( u n + 1 + u n + u n - 1 ) ⁢ u n = n - 2 ⁢ z ⁢ u n subscript 𝑢 𝑛 1 subscript 𝑢 𝑛 subscript 𝑢 𝑛 1 subscript 𝑢 𝑛 𝑛 2 𝑧 subscript 𝑢 𝑛 {\displaystyle{\displaystyle(u_{n+1}+u_{n}+u_{n-1})u_{n}=n-2zu_{n}}}
(u_{n+1}+u_{n}+u_{n-1})u_{n} = n-2zu_{n}		 

(u[n + 1]+ u[n]+ u[n - 1])*u[n] = n - 2*z*u[n]
 
(Subscript[u, n + 1]+ Subscript[u, n]+ Subscript[u, n - 1])*Subscript[u, n] == n - 2*z*Subscript[u, n]
 Skipped - no semantic math Skipped - no semantic math - -
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