View source for 13.23

{{DISPLAYTITLE:Confluent Hypergeometric Functions - 13.23 Integrals}}

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! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
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! 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/13.23.E1 13.23.E1] || [[Item:Q4635|<math>\int_{0}^{\infty}e^{-zt}t^{\nu-1}\WhittakerconfhyperM{\kappa}{\mu}@{t}\diff{t} = \frac{\EulerGamma@{\mu+\nu+\tfrac{1}{2}}}{\left(z+\frac{1}{2}\right)^{\mu+\nu+\frac{1}{2}}}\*\genhyperF{2}{1}@@{\tfrac{1}{2}+\mu-\kappa,\tfrac{1}{2}+\mu+\nu}{1+2\mu}{\frac{1}{z+\frac{1}{2}}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-zt}t^{\nu-1}\WhittakerconfhyperM{\kappa}{\mu}@{t}\diff{t} = \frac{\EulerGamma@{\mu+\nu+\tfrac{1}{2}}}{\left(z+\frac{1}{2}\right)^{\mu+\nu+\frac{1}{2}}}\*\genhyperF{2}{1}@@{\tfrac{1}{2}+\mu-\kappa,\tfrac{1}{2}+\mu+\nu}{1+2\mu}{\frac{1}{z+\frac{1}{2}}}</syntaxhighlight> || <math>\realpart@@{\mu+\nu+\tfrac{1}{2}} > 0, \realpart@@{z} > \tfrac{1}{2}, \realpart@@{(\mu+\nu+\tfrac{1}{2})} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(- z*t)*(t)^(nu - 1)* WhittakerM(kappa, mu, t), t = 0..infinity) = (GAMMA(mu + nu +(1)/(2)))/((z +(1)/(2))^(mu + nu +(1)/(2)))* hypergeom([(1)/(2)+ mu - kappa ,(1)/(2)+ mu + nu], [1 + 2*mu], (1)/(z +(1)/(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- z*t]*(t)^(\[Nu]- 1)* WhittakerM[\[Kappa], \[Mu], t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Mu]+ \[Nu]+Divide[1,2]],(z +Divide[1,2])^(\[Mu]+ \[Nu]+Divide[1,2])]* HypergeometricPFQ[{Divide[1,2]+ \[Mu]- \[Kappa],Divide[1,2]+ \[Mu]+ \[Nu]}, {1 + 2*\[Mu]}, Divide[1,z +Divide[1,2]]]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/13.23.E2 13.23.E2] || [[Item:Q4636|<math>\int_{0}^{\infty}e^{-zt}t^{\mu-\frac{1}{2}}\WhittakerconfhyperM{\kappa}{\mu}@{t}\diff{t} = \EulerGamma@{2\mu+1}\left(z+\tfrac{1}{2}\right)^{-\kappa-\mu-\frac{1}{2}}\*\left(z-\tfrac{1}{2}\right)^{\kappa-\mu-\frac{1}{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-zt}t^{\mu-\frac{1}{2}}\WhittakerconfhyperM{\kappa}{\mu}@{t}\diff{t} = \EulerGamma@{2\mu+1}\left(z+\tfrac{1}{2}\right)^{-\kappa-\mu-\frac{1}{2}}\*\left(z-\tfrac{1}{2}\right)^{\kappa-\mu-\frac{1}{2}}</syntaxhighlight> || <math>\realpart@@{\mu} > -\tfrac{1}{2}, \realpart@@{z} > \tfrac{1}{2}, \realpart@@{(2\mu+1)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(- z*t)*(t)^(mu -(1)/(2))* WhittakerM(kappa, mu, t), t = 0..infinity) = GAMMA(2*mu + 1)*(z +(1)/(2))^(- kappa - mu -(1)/(2))*(z -(1)/(2))^(kappa - mu -(1)/(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- z*t]*(t)^(\[Mu]-Divide[1,2])* WhittakerM[\[Kappa], \[Mu], t], {t, 0, Infinity}, GenerateConditions->None] == Gamma[2*\[Mu]+ 1]*(z +Divide[1,2])^(- \[Kappa]- \[Mu]-Divide[1,2])*(z -Divide[1,2])^(\[Kappa]- \[Mu]-Divide[1,2])</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/13.23.E3 13.23.E3] || [[Item:Q4637|<math>\frac{1}{\EulerGamma@{1+2\mu}}\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\nu-1}\WhittakerconfhyperM{\kappa}{\mu}@{t}\diff{t} = \frac{\EulerGamma@{\mu+\nu+\frac{1}{2}}\EulerGamma@{\kappa-\nu}}{\EulerGamma@{\frac{1}{2}+\mu+\kappa}\EulerGamma@{\frac{1}{2}+\mu-\nu}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\EulerGamma@{1+2\mu}}\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\nu-1}\WhittakerconfhyperM{\kappa}{\mu}@{t}\diff{t} = \frac{\EulerGamma@{\mu+\nu+\frac{1}{2}}\EulerGamma@{\kappa-\nu}}{\EulerGamma@{\frac{1}{2}+\mu+\kappa}\EulerGamma@{\frac{1}{2}+\mu-\nu}}</syntaxhighlight> || <math>-\tfrac{1}{2}-\realpart@@{\mu} < \realpart@@{\nu}, \realpart@@{\nu} < \realpart@@{\kappa}, \realpart@@{(1+2\mu)} > 0, \realpart@@{(\mu+\nu+\frac{1}{2})} > 0, \realpart@@{(\kappa-\nu)} > 0, \realpart@@{(\frac{1}{2}+\mu+\kappa)} > 0, \realpart@@{(\frac{1}{2}+\mu-\nu)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(GAMMA(1 + 2*mu))*int(exp(-(1)/(2)*t)*(t)^(nu - 1)* WhittakerM(kappa, mu, t), t = 0..infinity) = (GAMMA(mu + nu +(1)/(2))*GAMMA(kappa - nu))/(GAMMA((1)/(2)+ mu + kappa)*GAMMA((1)/(2)+ mu - nu))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,Gamma[1 + 2*\[Mu]]]*Integrate[Exp[-Divide[1,2]*t]*(t)^(\[Nu]- 1)* WhittakerM[\[Kappa], \[Mu], t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Mu]+ \[Nu]+Divide[1,2]]*Gamma[\[Kappa]- \[Nu]],Gamma[Divide[1,2]+ \[Mu]+ \[Kappa]]*Gamma[Divide[1,2]+ \[Mu]- \[Nu]]]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/13.23.E4 13.23.E4] || [[Item:Q4638|<math>\int_{0}^{\infty}e^{-zt}t^{\nu-1}\WhittakerconfhyperW{\kappa}{\mu}@{t}\diff{t} = \EulerGamma@{\tfrac{1}{2}+\mu+\nu}\EulerGamma@{\tfrac{1}{2}-\mu+\nu}\*\genhyperOlverF{2}{1}@@{\tfrac{1}{2}-\mu+\nu,\tfrac{1}{2}+\mu+\nu}{\nu-\kappa+1}{\tfrac{1}{2}-z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-zt}t^{\nu-1}\WhittakerconfhyperW{\kappa}{\mu}@{t}\diff{t} = \EulerGamma@{\tfrac{1}{2}+\mu+\nu}\EulerGamma@{\tfrac{1}{2}-\mu+\nu}\*\genhyperOlverF{2}{1}@@{\tfrac{1}{2}-\mu+\nu,\tfrac{1}{2}+\mu+\nu}{\nu-\kappa+1}{\tfrac{1}{2}-z}</syntaxhighlight> || <math>\realpart@{\nu+\tfrac{1}{2}} > |\realpart@@{\mu}|, \realpart@@{z} > -\tfrac{1}{2}, \realpart@@{(\tfrac{1}{2}+\mu+\nu)} > 0, \realpart@@{(\tfrac{1}{2}-\mu+\nu)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(- z*t)*(t)^(nu - 1)* WhittakerW(kappa, mu, t), t = 0..infinity) = GAMMA((1)/(2)+ mu + nu)*GAMMA((1)/(2)- mu + nu)* hypergeom([(1)/(2)- mu + nu ,(1)/(2)+ mu + nu], [nu - kappa + 1], (1)/(2)- z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[- z*t]*(t)^(\[Nu]- 1)* WhittakerW[\[Kappa], \[Mu], t], {t, 0, Infinity}, GenerateConditions->None] == Gamma[Divide[1,2]+ \[Mu]+ \[Nu]]*Gamma[Divide[1,2]- \[Mu]+ \[Nu]]* HypergeometricPFQRegularized[{Divide[1,2]- \[Mu]+ \[Nu],Divide[1,2]+ \[Mu]+ \[Nu]}, {\[Nu]- \[Kappa]+ 1}, Divide[1,2]- z]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [276 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Float(infinity)+Float(infinity)*I
Test Values: {kappa = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, z = 1/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .2394973555+.5504747838e-1*I
Test Values: {kappa = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, nu = 1/2-1/2*I*3^(1/2), z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
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| [https://dlmf.nist.gov/13.23.E5 13.23.E5] || [[Item:Q4639|<math>\int_{0}^{\infty}e^{\frac{1}{2}t}t^{\nu-1}\WhittakerconfhyperW{\kappa}{\mu}@{t}\diff{t} = \frac{\EulerGamma@{\frac{1}{2}+\mu+\nu}\EulerGamma@{\frac{1}{2}-\mu+\nu}\EulerGamma@{-\kappa-\nu}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{\frac{1}{2}t}t^{\nu-1}\WhittakerconfhyperW{\kappa}{\mu}@{t}\diff{t} = \frac{\EulerGamma@{\frac{1}{2}+\mu+\nu}\EulerGamma@{\frac{1}{2}-\mu+\nu}\EulerGamma@{-\kappa-\nu}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}</syntaxhighlight> || <math>|\realpart@@{\mu}|-\tfrac{1}{2} < \realpart@@{\nu}, \realpart@@{\nu} < -\realpart@@{\kappa}, \realpart@@{(\frac{1}{2}+\mu+\nu)} > 0, \realpart@@{(\frac{1}{2}-\mu+\nu)} > 0, \realpart@@{(-\kappa-\nu)} > 0, \realpart@@{(\frac{1}{2}+\mu-\kappa)} > 0, \realpart@@{(\frac{1}{2}-\mu-\kappa)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp((1)/(2)*t)*(t)^(nu - 1)* WhittakerW(kappa, mu, t), t = 0..infinity) = (GAMMA((1)/(2)+ mu + nu)*GAMMA((1)/(2)- mu + nu)*GAMMA(- kappa - nu))/(GAMMA((1)/(2)+ mu - kappa)*GAMMA((1)/(2)- mu - kappa))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[Divide[1,2]*t]*(t)^(\[Nu]- 1)* WhittakerW[\[Kappa], \[Mu], t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[Divide[1,2]+ \[Mu]+ \[Nu]]*Gamma[Divide[1,2]- \[Mu]+ \[Nu]]*Gamma[- \[Kappa]- \[Nu]],Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]*Gamma[Divide[1,2]- \[Mu]- \[Kappa]]]</syntaxhighlight> || Failure || Aborted || Manual Skip! || Successful [Tested: 56]
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| [https://dlmf.nist.gov/13.23.E6 13.23.E6] || [[Item:Q4640|<math>\frac{1}{\EulerGamma@{1+2\mu}2\pi\iunit}\int_{-\infty}^{(0+)}e^{zt+\frac{1}{2}t^{-1}}t^{\kappa}\WhittakerconfhyperM{\kappa}{\mu}@{t^{-1}}\diff{t} = \frac{z^{-\kappa-\frac{1}{2}}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\modBesselI{2\mu}@{2\sqrt{z}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\EulerGamma@{1+2\mu}2\pi\iunit}\int_{-\infty}^{(0+)}e^{zt+\frac{1}{2}t^{-1}}t^{\kappa}\WhittakerconfhyperM{\kappa}{\mu}@{t^{-1}}\diff{t} = \frac{z^{-\kappa-\frac{1}{2}}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\modBesselI{2\mu}@{2\sqrt{z}}</syntaxhighlight> || <math>\realpart@@{z} > 0, \realpart@@{(1+2\mu)} > 0, \realpart@@{(\frac{1}{2}+\mu-\kappa)} > 0, \realpart@@{((2\mu)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(GAMMA(1 + 2*mu)*2*Pi*I)*int(exp(z*t +(1)/(2)*(t)^(- 1))*(t)^(kappa)* WhittakerM(kappa, mu, (t)^(- 1)), t = - infinity..(0 +)) = ((z)^(- kappa -(1)/(2)))/(GAMMA((1)/(2)+ mu - kappa))*BesselI(2*mu, 2*sqrt(z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,Gamma[1 + 2*\[Mu]]*2*Pi*I]*Integrate[Exp[z*t +Divide[1,2]*(t)^(- 1)]*(t)^\[Kappa]* WhittakerM[\[Kappa], \[Mu], (t)^(- 1)], {t, - Infinity, (0 +)}, GenerateConditions->None] == Divide[(z)^(- \[Kappa]-Divide[1,2]),Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]]*BesselI[2*\[Mu], 2*Sqrt[z]]</syntaxhighlight> || Error || Failure || - || Error
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| [https://dlmf.nist.gov/13.23.E7 13.23.E7] || [[Item:Q4641|<math>\frac{1}{2\pi\iunit}\int_{-\infty}^{(0+)}e^{zt+\frac{1}{2}t^{-1}}t^{\kappa}\WhittakerconfhyperW{\kappa}{\mu}@{t^{-1}}\diff{t} = \frac{2z^{-\kappa-\frac{1}{2}}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}\modBesselK{2\mu}@{2\sqrt{z}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{2\pi\iunit}\int_{-\infty}^{(0+)}e^{zt+\frac{1}{2}t^{-1}}t^{\kappa}\WhittakerconfhyperW{\kappa}{\mu}@{t^{-1}}\diff{t} = \frac{2z^{-\kappa-\frac{1}{2}}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}\modBesselK{2\mu}@{2\sqrt{z}}</syntaxhighlight> || <math>\realpart@@{z} > 0, \realpart@@{(\frac{1}{2}+\mu-\kappa)} > 0, \realpart@@{(\frac{1}{2}-\mu-\kappa)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(2*Pi*I)*int(exp(z*t +(1)/(2)*(t)^(- 1))*(t)^(kappa)* WhittakerW(kappa, mu, (t)^(- 1)), t = - infinity..(0 +)) = (2*(z)^(- kappa -(1)/(2)))/(GAMMA((1)/(2)+ mu - kappa)*GAMMA((1)/(2)- mu - kappa))*BesselK(2*mu, 2*sqrt(z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2*Pi*I]*Integrate[Exp[z*t +Divide[1,2]*(t)^(- 1)]*(t)^\[Kappa]* WhittakerW[\[Kappa], \[Mu], (t)^(- 1)], {t, - Infinity, (0 +)}, GenerateConditions->None] == Divide[2*(z)^(- \[Kappa]-Divide[1,2]),Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]*Gamma[Divide[1,2]- \[Mu]- \[Kappa]]]*BesselK[2*\[Mu], 2*Sqrt[z]]</syntaxhighlight> || Error || Failure || - || Error
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| [https://dlmf.nist.gov/13.23.E8 13.23.E8] || [[Item:Q4642|<math>\frac{1}{\EulerGamma@{1+2\mu}}\int_{0}^{\infty}\cos@{2xt}e^{-\frac{1}{2}t^{2}}t^{-2\mu-1}\WhittakerconfhyperM{\kappa}{\mu}@{t^{2}}\diff{t} = \frac{\sqrt{\pi}e^{-\frac{1}{2}x^{2}}x^{\mu+\kappa-1}}{2\EulerGamma@{\frac{1}{2}+\mu+\kappa}}\WhittakerconfhyperW{\frac{1}{2}\kappa-\frac{3}{2}\mu}{\frac{1}{2}\kappa+\frac{1}{2}\mu}@{x^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\EulerGamma@{1+2\mu}}\int_{0}^{\infty}\cos@{2xt}e^{-\frac{1}{2}t^{2}}t^{-2\mu-1}\WhittakerconfhyperM{\kappa}{\mu}@{t^{2}}\diff{t} = \frac{\sqrt{\pi}e^{-\frac{1}{2}x^{2}}x^{\mu+\kappa-1}}{2\EulerGamma@{\frac{1}{2}+\mu+\kappa}}\WhittakerconfhyperW{\frac{1}{2}\kappa-\frac{3}{2}\mu}{\frac{1}{2}\kappa+\frac{1}{2}\mu}@{x^{2}}</syntaxhighlight> || <math>\realpart@{\kappa+\mu} > -\tfrac{1}{2}, \realpart@@{(1+2\mu)} > 0, \realpart@@{(\frac{1}{2}+\mu+\kappa)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(GAMMA(1 + 2*mu))*int(cos(2*x*t)*exp(-(1)/(2)*(t)^(2))*(t)^(- 2*mu - 1)* WhittakerM(kappa, mu, (t)^(2)), t = 0..infinity) = (sqrt(Pi)*exp(-(1)/(2)*(x)^(2))*(x)^(mu + kappa - 1))/(2*GAMMA((1)/(2)+ mu + kappa))*WhittakerW((1)/(2)*kappa -(3)/(2)*mu, (1)/(2)*kappa +(1)/(2)*mu, (x)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,Gamma[1 + 2*\[Mu]]]*Integrate[Cos[2*x*t]*Exp[-Divide[1,2]*(t)^(2)]*(t)^(- 2*\[Mu]- 1)* WhittakerM[\[Kappa], \[Mu], (t)^(2)], {t, 0, Infinity}, GenerateConditions->None] == Divide[Sqrt[Pi]*Exp[-Divide[1,2]*(x)^(2)]*(x)^(\[Mu]+ \[Kappa]- 1),2*Gamma[Divide[1,2]+ \[Mu]+ \[Kappa]]]*WhittakerW[Divide[1,2]*\[Kappa]-Divide[3,2]*\[Mu], Divide[1,2]*\[Kappa]+Divide[1,2]*\[Mu], (x)^(2)]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/13.23.E9 13.23.E9] || [[Item:Q4643|<math>\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\mu-\frac{1}{2}(\nu+1)}\WhittakerconfhyperM{\kappa}{\mu}@{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \frac{\EulerGamma@{1+2\mu}}{\EulerGamma@{\frac{1}{2}-\mu+\kappa+\nu}}\*e^{-\frac{1}{2}x}x^{\frac{1}{2}(\kappa-\mu-\frac{3}{2})}\*\WhittakerconfhyperM{\frac{1}{2}(\kappa+3\mu-\nu+\frac{1}{2})}{\frac{1}{2}(\kappa-\mu+\nu-\frac{1}{2})}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\mu-\frac{1}{2}(\nu+1)}\WhittakerconfhyperM{\kappa}{\mu}@{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \frac{\EulerGamma@{1+2\mu}}{\EulerGamma@{\frac{1}{2}-\mu+\kappa+\nu}}\*e^{-\frac{1}{2}x}x^{\frac{1}{2}(\kappa-\mu-\frac{3}{2})}\*\WhittakerconfhyperM{\frac{1}{2}(\kappa+3\mu-\nu+\frac{1}{2})}{\frac{1}{2}(\kappa-\mu+\nu-\frac{1}{2})}@{x}</syntaxhighlight> || <math>x > 0, -\tfrac{1}{2} < \realpart@@{\mu}, \realpart@@{\mu} < \realpart@{\kappa+\tfrac{1}{2}\nu}+\tfrac{3}{4}, \realpart@@{(\nu+k+1)} > 0, \realpart@@{(1+2\mu)} > 0, \realpart@@{(\frac{1}{2}-\mu+\kappa+\nu)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(-(1)/(2)*t)*(t)^(mu -(1)/(2)*(nu + 1))* WhittakerM(kappa, mu, t)*BesselJ(nu, 2*sqrt(x*t)), t = 0..infinity) = (GAMMA(1 + 2*mu))/(GAMMA((1)/(2)- mu + kappa + nu))* exp(-(1)/(2)*x)*(x)^((1)/(2)*(kappa - mu -(3)/(2)))* WhittakerM((1)/(2)*(kappa + 3*mu - nu +(1)/(2)), (1)/(2)*(kappa - mu + nu -(1)/(2)), x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[-Divide[1,2]*t]*(t)^(\[Mu]-Divide[1,2]*(\[Nu]+ 1))* WhittakerM[\[Kappa], \[Mu], t]*BesselJ[\[Nu], 2*Sqrt[x*t]], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[1 + 2*\[Mu]],Gamma[Divide[1,2]- \[Mu]+ \[Kappa]+ \[Nu]]]* Exp[-Divide[1,2]*x]*(x)^(Divide[1,2]*(\[Kappa]- \[Mu]-Divide[3,2]))* WhittakerM[Divide[1,2]*(\[Kappa]+ 3*\[Mu]- \[Nu]+Divide[1,2]), Divide[1,2]*(\[Kappa]- \[Mu]+ \[Nu]-Divide[1,2]), x]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/13.23.E10 13.23.E10] || [[Item:Q4644|<math>\frac{1}{\EulerGamma@{1+2\mu}}\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\frac{1}{2}(\nu-1)-\mu}\WhittakerconfhyperM{\kappa}{\mu}@{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \frac{e^{-\frac{1}{2}x}x^{\frac{1}{2}(\kappa+\mu-\frac{3}{2})}}{\EulerGamma@{\frac{1}{2}+\mu+\kappa}}\*\WhittakerconfhyperW{\frac{1}{2}(\kappa-3\mu+\nu+\frac{1}{2})}{\frac{1}{2}(\kappa+\mu-\nu-\frac{1}{2})}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\EulerGamma@{1+2\mu}}\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\frac{1}{2}(\nu-1)-\mu}\WhittakerconfhyperM{\kappa}{\mu}@{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \frac{e^{-\frac{1}{2}x}x^{\frac{1}{2}(\kappa+\mu-\frac{3}{2})}}{\EulerGamma@{\frac{1}{2}+\mu+\kappa}}\*\WhittakerconfhyperW{\frac{1}{2}(\kappa-3\mu+\nu+\frac{1}{2})}{\frac{1}{2}(\kappa+\mu-\nu-\frac{1}{2})}@{x}</syntaxhighlight> || <math>x > 0, -1 < \realpart@@{\nu}, \realpart@@{\nu} < 2\realpart@{\mu+\kappa}+\tfrac{1}{2}, \realpart@@{(\nu+k+1)} > 0, \realpart@@{(1+2\mu)} > 0, \realpart@@{(\frac{1}{2}+\mu+\kappa)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(GAMMA(1 + 2*mu))*int(exp(-(1)/(2)*t)*(t)^((1)/(2)*(nu - 1)- mu)* WhittakerM(kappa, mu, t)*BesselJ(nu, 2*sqrt(x*t)), t = 0..infinity) = (exp(-(1)/(2)*x)*(x)^((1)/(2)*(kappa + mu -(3)/(2))))/(GAMMA((1)/(2)+ mu + kappa))* WhittakerW((1)/(2)*(kappa - 3*mu + nu +(1)/(2)), (1)/(2)*(kappa + mu - nu -(1)/(2)), x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,Gamma[1 + 2*\[Mu]]]*Integrate[Exp[-Divide[1,2]*t]*(t)^(Divide[1,2]*(\[Nu]- 1)- \[Mu])* WhittakerM[\[Kappa], \[Mu], t]*BesselJ[\[Nu], 2*Sqrt[x*t]], {t, 0, Infinity}, GenerateConditions->None] == Divide[Exp[-Divide[1,2]*x]*(x)^(Divide[1,2]*(\[Kappa]+ \[Mu]-Divide[3,2])),Gamma[Divide[1,2]+ \[Mu]+ \[Kappa]]]* WhittakerW[Divide[1,2]*(\[Kappa]- 3*\[Mu]+ \[Nu]+Divide[1,2]), Divide[1,2]*(\[Kappa]+ \[Mu]- \[Nu]-Divide[1,2]), x]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/13.23.E11 13.23.E11] || [[Item:Q4645|<math>\int_{0}^{\infty}e^{\frac{1}{2}t}t^{\frac{1}{2}(\nu-1)-\mu}\WhittakerconfhyperW{\kappa}{\mu}@{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \frac{\EulerGamma@{\nu-2\mu+1}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*e^{\frac{1}{2}x}x^{\frac{1}{2}(\mu-\kappa-\frac{3}{2})}\*\WhittakerconfhyperW{\frac{1}{2}(\kappa+3\mu-\nu-\frac{1}{2})}{\frac{1}{2}(\kappa-\mu+\nu+\frac{1}{2})}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{\frac{1}{2}t}t^{\frac{1}{2}(\nu-1)-\mu}\WhittakerconfhyperW{\kappa}{\mu}@{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \frac{\EulerGamma@{\nu-2\mu+1}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*e^{\frac{1}{2}x}x^{\frac{1}{2}(\mu-\kappa-\frac{3}{2})}\*\WhittakerconfhyperW{\frac{1}{2}(\kappa+3\mu-\nu-\frac{1}{2})}{\frac{1}{2}(\kappa-\mu+\nu+\frac{1}{2})}@{x}</syntaxhighlight> || <math>x > 0, \max(2\realpart@@{\mu}-1 < \realpart@@{\nu}, -1) < \realpart@@{\nu}, \realpart@@{\nu} < 2\realpart@@{\mu-\kappa}+\tfrac{3}{2}, \realpart@@{(\nu+k+1)} > 0, \realpart@@{(\nu-2\mu+1)} > 0, \realpart@@{(\frac{1}{2}+\mu-\kappa)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp((1)/(2)*t)*(t)^((1)/(2)*(nu - 1)- mu)* WhittakerW(kappa, mu, t)*BesselJ(nu, 2*sqrt(x*t)), t = 0..infinity) = (GAMMA(nu - 2*mu + 1))/(GAMMA((1)/(2)+ mu - kappa))* exp((1)/(2)*x)*(x)^((1)/(2)*(mu - kappa -(3)/(2)))* WhittakerW((1)/(2)*(kappa + 3*mu - nu -(1)/(2)), (1)/(2)*(kappa - mu + nu +(1)/(2)), x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[Divide[1,2]*t]*(t)^(Divide[1,2]*(\[Nu]- 1)- \[Mu])* WhittakerW[\[Kappa], \[Mu], t]*BesselJ[\[Nu], 2*Sqrt[x*t]], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Nu]- 2*\[Mu]+ 1],Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]]* Exp[Divide[1,2]*x]*(x)^(Divide[1,2]*(\[Mu]- \[Kappa]-Divide[3,2]))* WhittakerW[Divide[1,2]*(\[Kappa]+ 3*\[Mu]- \[Nu]-Divide[1,2]), Divide[1,2]*(\[Kappa]- \[Mu]+ \[Nu]+Divide[1,2]), x]</syntaxhighlight> || Failure || Aborted || Manual Skip! || Skipped - Because timed out
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| [https://dlmf.nist.gov/13.23.E12 13.23.E12] || [[Item:Q4646|<math>\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\frac{1}{2}(\nu-1)-\mu}\WhittakerconfhyperW{\kappa}{\mu}@{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \frac{\EulerGamma@{\nu-2\mu+1}}{\EulerGamma@{\frac{3}{2}-\mu-\kappa+\nu}}\*e^{-\frac{1}{2}x}x^{\frac{1}{2}(\mu+\kappa-\frac{3}{2})}\*\WhittakerconfhyperM{\frac{1}{2}(\kappa-3\mu+\nu+\frac{1}{2})}{\frac{1}{2}(\nu-\mu-\kappa+\frac{1}{2})}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\frac{1}{2}(\nu-1)-\mu}\WhittakerconfhyperW{\kappa}{\mu}@{t}\BesselJ{\nu}@{2\sqrt{xt}}\diff{t} = \frac{\EulerGamma@{\nu-2\mu+1}}{\EulerGamma@{\frac{3}{2}-\mu-\kappa+\nu}}\*e^{-\frac{1}{2}x}x^{\frac{1}{2}(\mu+\kappa-\frac{3}{2})}\*\WhittakerconfhyperM{\frac{1}{2}(\kappa-3\mu+\nu+\frac{1}{2})}{\frac{1}{2}(\nu-\mu-\kappa+\frac{1}{2})}@{x}</syntaxhighlight> || <math>x > 0, \max(2\realpart@@{\mu}-1 < \realpart@@{\nu}, -1) < \realpart@@{\nu}, \realpart@@{(\nu+k+1)} > 0, \realpart@@{(\nu-2\mu+1)} > 0, \realpart@@{(\frac{3}{2}-\mu-\kappa+\nu)} > 0</math> || <syntaxhighlight lang=mathematica>int(exp(-(1)/(2)*t)*(t)^((1)/(2)*(nu - 1)- mu)* WhittakerW(kappa, mu, t)*BesselJ(nu, 2*sqrt(x*t)), t = 0..infinity) = (GAMMA(nu - 2*mu + 1))/(GAMMA((3)/(2)- mu - kappa + nu))* exp(-(1)/(2)*x)*(x)^((1)/(2)*(mu + kappa -(3)/(2)))* WhittakerM((1)/(2)*(kappa - 3*mu + nu +(1)/(2)), (1)/(2)*(nu - mu - kappa +(1)/(2)), x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Integrate[Exp[-Divide[1,2]*t]*(t)^(Divide[1,2]*(\[Nu]- 1)- \[Mu])* WhittakerW[\[Kappa], \[Mu], t]*BesselJ[\[Nu], 2*Sqrt[x*t]], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Nu]- 2*\[Mu]+ 1],Gamma[Divide[3,2]- \[Mu]- \[Kappa]+ \[Nu]]]* Exp[-Divide[1,2]*x]*(x)^(Divide[1,2]*(\[Mu]+ \[Kappa]-Divide[3,2]))* WhittakerM[Divide[1,2]*(\[Kappa]- 3*\[Mu]+ \[Nu]+Divide[1,2]), Divide[1,2]*(\[Nu]- \[Mu]- \[Kappa]+Divide[1,2]), x]</syntaxhighlight> || Failure || Aborted || Manual Skip! || Skipped - Because timed out
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