View source for 10.54

{{DISPLAYTITLE:Bessel Functions - 10.54 Integral Representations}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/10.54.E1 10.54.E1] || <math qid="Q3759">\sphBesselJ{n}@{z} = \frac{z^{n}}{2^{n+1}n!}\int_{0}^{\pi}\cos@{z\cos@@{\theta}}(\sin@@{\theta})^{2n+1}\diff{\theta}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{n}@{z} = \frac{z^{n}}{2^{n+1}n!}\int_{0}^{\pi}\cos@{z\cos@@{\theta}}(\sin@@{\theta})^{2n+1}\diff{\theta}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselJ[n, z] == Divide[(z)^(n),(2)^(n + 1)* (n)!]*Integrate[Cos[z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*n + 1), {\[Theta], 0, Pi}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 21]
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| [https://dlmf.nist.gov/10.54.E2 10.54.E2] || <math qid="Q3760">\sphBesselJ{n}@{z} = \frac{(-i)^{n}}{2}\int_{0}^{\pi}e^{iz\cos@@{\theta}}\assLegendreP[]{n}@{\cos@@{\theta}}\sin@@{\theta}\diff{\theta}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{n}@{z} = \frac{(-i)^{n}}{2}\int_{0}^{\pi}e^{iz\cos@@{\theta}}\assLegendreP[]{n}@{\cos@@{\theta}}\sin@@{\theta}\diff{\theta}</syntaxhighlight> || <math>\realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselJ[n, z] == Divide[(- I)^(n),2]*Integrate[Exp[I*z*Cos[\[Theta]]]*LegendreP[n, 0, 3, Cos[\[Theta]]]*Sin[\[Theta]], {\[Theta], 0, Pi}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Aborted || - || Successful [Tested: 21]
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| [https://dlmf.nist.gov/10.54.E3 10.54.E3] || <math qid="Q3761">\modsphBesselK{n}@{z} = \frac{\pi}{2}\int_{1}^{\infty}e^{-zt}\assLegendreP[]{n}@{t}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\modsphBesselK{n}@{z} = \frac{\pi}{2}\int_{1}^{\infty}e^{-zt}\assLegendreP[]{n}@{t}\diff{t}</syntaxhighlight> || <math>|\phase@@{z}| < \tfrac{1}{2}\pi.</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[1/2 Pi /z] BesselK[n + 1/2, z] == Divide[Pi,2]*Integrate[Exp[- z*t]*LegendreP[n, 0, 3, t], {t, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Aborted || - || Skipped - Because timed out
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| [https://dlmf.nist.gov/10.54.E4 10.54.E4] || <math qid="Q3762">\sphBesselJ{n}@{z} = \frac{(-i)^{n+1}}{2\pi}\int_{i\infty}^{(-1+,1+)}e^{izt}\assLegendreQ[]{n}@{t}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphBesselJ{n}@{z} = \frac{(-i)^{n+1}}{2\pi}\int_{i\infty}^{(-1+,1+)}e^{izt}\assLegendreQ[]{n}@{t}\diff{t}</syntaxhighlight> || <math>|\phase@@{z}| < \tfrac{1}{2}\pi., \realpart@@{((n+\frac{1}{2})+k+1)} > 0, \realpart@@{((-n-\frac{1}{2})+k+1)} > 0, \realpart@@{((-(-n-\frac{1}{2}))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalBesselJ[n, z] == Divide[(- I)^(n + 1),2*Pi]*Integrate[Exp[I*z*t]*LegendreQ[n, 0, 3, t], {t, I*Infinity, (- 1 + , 1 +)}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Error
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| [https://dlmf.nist.gov/10.54#Ex1 10.54#Ex1] || <math qid="Q3763">\sphHankelh{1}{n}@{z} = \frac{(-i)^{n+1}}{\pi}\int_{i\infty}^{(1+)}e^{izt}\assLegendreQ[]{n}@{t}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphHankelh{1}{n}@{z} = \frac{(-i)^{n+1}}{\pi}\int_{i\infty}^{(1+)}e^{izt}\assLegendreQ[]{n}@{t}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalHankelH1[n, z] == Divide[(- I)^(n + 1),Pi]*Integrate[Exp[I*z*t]*LegendreQ[n, 0, 3, t], {t, I*Infinity, (1 +)}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Error
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| [https://dlmf.nist.gov/10.54#Ex2 10.54#Ex2] || <math qid="Q3764">\sphHankelh{2}{n}@{z} = \frac{(-i)^{n+1}}{\pi}\int_{i\infty}^{(-1+)}e^{izt}\assLegendreQ[]{n}@{t}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sphHankelh{2}{n}@{z} = \frac{(-i)^{n+1}}{\pi}\int_{i\infty}^{(-1+)}e^{izt}\assLegendreQ[]{n}@{t}\diff{t}</syntaxhighlight> || <math>|\phase@@{z}| < \tfrac{1}{2}\pi.</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>SphericalHankelH2[n, z] == Divide[(- I)^(n + 1),Pi]*Integrate[Exp[I*z*t]*LegendreQ[n, 0, 3, t], {t, I*Infinity, (- 1 +)}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Error
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