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Admin moved page Main Page to Verifying DLMF with Maple and Mathematica |
Admin moved page Main Page to Verifying DLMF with Maple and Mathematica |
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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/13.16.E1 13.16.E1] | | | [https://dlmf.nist.gov/13.16.E1 13.16.E1] || <math qid="Q4552">\WhittakerconfhyperM{\kappa}{\mu}@{z} = \frac{\EulerGamma@{1+2\mu}z^{\mu+\frac{1}{2}}2^{-2\mu}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}+\mu+\kappa}}\*\int_{-1}^{1}e^{\frac{1}{2}zt}(1+t)^{\mu-\frac{1}{2}-\kappa}(1-t)^{\mu-\frac{1}{2}+\kappa}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperM{\kappa}{\mu}@{z} = \frac{\EulerGamma@{1+2\mu}z^{\mu+\frac{1}{2}}2^{-2\mu}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}+\mu+\kappa}}\*\int_{-1}^{1}e^{\frac{1}{2}zt}(1+t)^{\mu-\frac{1}{2}-\kappa}(1-t)^{\mu-\frac{1}{2}+\kappa}\diff{t}</syntaxhighlight> || <math>\realpart@@{\mu}+\tfrac{1}{2} > \left|\realpart@@{\kappa}\right|, \realpart@@{(1+2\mu)} > 0, \realpart@@{(\frac{1}{2}+\mu-\kappa)} > 0, \realpart@@{(\frac{1}{2}+\mu+\kappa)} > 0</math> || <syntaxhighlight lang=mathematica>WhittakerM(kappa, mu, z) = (GAMMA(1 + 2*mu)*(z)^(mu +(1)/(2))* (2)^(- 2*mu))/(GAMMA((1)/(2)+ mu - kappa)*GAMMA((1)/(2)+ mu + kappa))* int(exp((1)/(2)*z*t)*(1 + t)^(mu -(1)/(2)- kappa)*(1 - t)^(mu -(1)/(2)+ kappa), t = - 1..1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerM[\[Kappa], \[Mu], z] == Divide[Gamma[1 + 2*\[Mu]]*(z)^(\[Mu]+Divide[1,2])* (2)^(- 2*\[Mu]),Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]*Gamma[Divide[1,2]+ \[Mu]+ \[Kappa]]]* Integrate[Exp[Divide[1,2]*z*t]*(1 + t)^(\[Mu]-Divide[1,2]- \[Kappa])*(1 - t)^(\[Mu]-Divide[1,2]+ \[Kappa]), {t, - 1, 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Skipped - Because timed out || Successful [Tested: 252] | ||
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| [https://dlmf.nist.gov/13.16.E2 13.16.E2] | | | [https://dlmf.nist.gov/13.16.E2 13.16.E2] || <math qid="Q4553">\WhittakerconfhyperM{\kappa}{\mu}@{z} = \frac{\EulerGamma@{1+2\mu}z^{\lambda}}{\EulerGamma@{1+2\mu-2\lambda}\EulerGamma@{2\lambda}}\*\int_{0}^{1}\WhittakerconfhyperM{\kappa-\lambda}{\mu-\lambda}@{zt}e^{\frac{1}{2}z(t-1)}t^{\mu-\lambda-\frac{1}{2}}{(1-t)^{2\lambda-1}}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperM{\kappa}{\mu}@{z} = \frac{\EulerGamma@{1+2\mu}z^{\lambda}}{\EulerGamma@{1+2\mu-2\lambda}\EulerGamma@{2\lambda}}\*\int_{0}^{1}\WhittakerconfhyperM{\kappa-\lambda}{\mu-\lambda}@{zt}e^{\frac{1}{2}z(t-1)}t^{\mu-\lambda-\frac{1}{2}}{(1-t)^{2\lambda-1}}\diff{t}</syntaxhighlight> || <math>\realpart@@{\mu}+\tfrac{1}{2} > \realpart@@{\lambda}, \realpart@@{\lambda} > 0, \realpart@@{(1+2\mu)} > 0, \realpart@@{(1+2\mu-2\lambda)} > 0, \realpart@@{(2\lambda)} > 0</math> || <syntaxhighlight lang=mathematica>WhittakerM(kappa, mu, z) = (GAMMA(1 + 2*mu)*(z)^(lambda))/(GAMMA(1 + 2*mu - 2*lambda)*GAMMA(2*lambda))* int(WhittakerM(kappa - lambda, mu - lambda, z*t)*exp((1)/(2)*z*(t - 1))*(t)^(mu - lambda -(1)/(2))*(1 - t)^(2*lambda - 1), t = 0..1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerM[\[Kappa], \[Mu], z] == Divide[Gamma[1 + 2*\[Mu]]*(z)^\[Lambda],Gamma[1 + 2*\[Mu]- 2*\[Lambda]]*Gamma[2*\[Lambda]]]* Integrate[WhittakerM[\[Kappa]- \[Lambda], \[Mu]- \[Lambda], z*t]*Exp[Divide[1,2]*z*(t - 1)]*(t)^(\[Mu]- \[Lambda]-Divide[1,2])*(1 - t)^(2*\[Lambda]- 1), {t, 0, 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out | ||
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| [https://dlmf.nist.gov/13.16.E3 13.16.E3] | | | [https://dlmf.nist.gov/13.16.E3 13.16.E3] || <math qid="Q4554">\frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{z} = \frac{\sqrt{z}e^{\frac{1}{2}z}}{\EulerGamma@{\frac{1}{2}+\mu+\kappa}}\int_{0}^{\infty}e^{-t}t^{\kappa-\frac{1}{2}}\BesselJ{2\mu}@{2\sqrt{zt}}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{z} = \frac{\sqrt{z}e^{\frac{1}{2}z}}{\EulerGamma@{\frac{1}{2}+\mu+\kappa}}\int_{0}^{\infty}e^{-t}t^{\kappa-\frac{1}{2}}\BesselJ{2\mu}@{2\sqrt{zt}}\diff{t}</syntaxhighlight> || <math>\realpart@{\kappa+\mu}+\tfrac{1}{2} > 0, \realpart@@{((2\mu)+k+1)} > 0, \realpart@@{(1+2\mu)} > 0, \realpart@@{(\frac{1}{2}+\mu+\kappa)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(GAMMA(1 + 2*mu))*WhittakerM(kappa, mu, z) = (sqrt(z)*exp((1)/(2)*z))/(GAMMA((1)/(2)+ mu + kappa))*int(exp(- t)*(t)^(kappa -(1)/(2))* BesselJ(2*mu, 2*sqrt(z*t)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,Gamma[1 + 2*\[Mu]]]*WhittakerM[\[Kappa], \[Mu], z] == Divide[Sqrt[z]*Exp[Divide[1,2]*z],Gamma[Divide[1,2]+ \[Mu]+ \[Kappa]]]*Integrate[Exp[- t]*(t)^(\[Kappa]-Divide[1,2])* BesselJ[2*\[Mu], 2*Sqrt[z*t]], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Aborted || - || Skipped - Because timed out | ||
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| [https://dlmf.nist.gov/13.16.E4 13.16.E4] | | | [https://dlmf.nist.gov/13.16.E4 13.16.E4] || <math qid="Q4555">\frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{z} = \frac{\sqrt{z}e^{-\frac{1}{2}z}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*\int_{0}^{\infty}e^{-t}t^{-\kappa-\frac{1}{2}}\modBesselI{2\mu}@{2\sqrt{zt}}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{z} = \frac{\sqrt{z}e^{-\frac{1}{2}z}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*\int_{0}^{\infty}e^{-t}t^{-\kappa-\frac{1}{2}}\modBesselI{2\mu}@{2\sqrt{zt}}\diff{t}</syntaxhighlight> || <math>\realpart@@{(\kappa-\mu)-\tfrac{1}{2}} < 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))*WhittakerM(kappa, mu, z) = (sqrt(z)*exp(-(1)/(2)*z))/(GAMMA((1)/(2)+ mu - kappa))* int(exp(- t)*(t)^(- kappa -(1)/(2))* BesselI(2*mu, 2*sqrt(z*t)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,Gamma[1 + 2*\[Mu]]]*WhittakerM[\[Kappa], \[Mu], z] == Divide[Sqrt[z]*Exp[-Divide[1,2]*z],Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]]* Integrate[Exp[- t]*(t)^(- \[Kappa]-Divide[1,2])* BesselI[2*\[Mu], 2*Sqrt[z*t]], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [42 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .5483729950e-2+.5411197480e-1*I | ||
Test Values: {kappa = -3/2, mu = 2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .2482822497e-1-.2550894001e-1*I | Test Values: {kappa = -3/2, mu = 2, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .2482822497e-1-.2550894001e-1*I | ||
Test Values: {kappa = -3/2, mu = 2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 300] | Test Values: {kappa = -3/2, mu = 2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 300] | ||
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| [https://dlmf.nist.gov/13.16.E5 13.16.E5] | | | [https://dlmf.nist.gov/13.16.E5 13.16.E5] || <math qid="Q4556">\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{z^{\mu+\frac{1}{2}}2^{-2\mu}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*\int_{1}^{\infty}e^{-\frac{1}{2}zt}(t-1)^{\mu-\frac{1}{2}-\kappa}(t+1)^{\mu-\frac{1}{2}+\kappa}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{z^{\mu+\frac{1}{2}}2^{-2\mu}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*\int_{1}^{\infty}e^{-\frac{1}{2}zt}(t-1)^{\mu-\frac{1}{2}-\kappa}(t+1)^{\mu-\frac{1}{2}+\kappa}\diff{t}</syntaxhighlight> || <math>\realpart@@{\mu}+\tfrac{1}{2} > \realpart@@{\kappa}, |\phase{z}| < \frac{1}{2}\pi, \realpart@@{(\frac{1}{2}+\mu-\kappa)} > 0</math> || <syntaxhighlight lang=mathematica>WhittakerW(kappa, mu, z) = ((z)^(mu +(1)/(2))* (2)^(- 2*mu))/(GAMMA((1)/(2)+ mu - kappa))* int(exp(-(1)/(2)*z*t)*(t - 1)^(mu -(1)/(2)- kappa)*(t + 1)^(mu -(1)/(2)+ kappa), t = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerW[\[Kappa], \[Mu], z] == Divide[(z)^(\[Mu]+Divide[1,2])* (2)^(- 2*\[Mu]),Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]]* Integrate[Exp[-Divide[1,2]*z*t]*(t - 1)^(\[Mu]-Divide[1,2]- \[Kappa])*(t + 1)^(\[Mu]-Divide[1,2]+ \[Kappa]), {t, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Successful [Tested: 300] | ||
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| [https://dlmf.nist.gov/13.16.E6 13.16.E6] | | | [https://dlmf.nist.gov/13.16.E6 13.16.E6] || <math qid="Q4557">\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{e^{-\frac{1}{2}z}z^{\kappa+1}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}\*\int_{0}^{\infty}\frac{\WhittakerconfhyperW{-\kappa}{\mu}@{t}e^{-\frac{1}{2}t}t^{-\kappa-1}}{t+z}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{e^{-\frac{1}{2}z}z^{\kappa+1}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}\*\int_{0}^{\infty}\frac{\WhittakerconfhyperW{-\kappa}{\mu}@{t}e^{-\frac{1}{2}t}t^{-\kappa-1}}{t+z}\diff{t}</syntaxhighlight> || <math>|\phase{z}| < \pi, \realpart@{\frac{1}{2}+\mu-\kappa} > \max\left(2\realpart@@{\mu}, \realpart@@{(\frac{1}{2}+\mu-\kappa)} > 0, \realpart@@{(\frac{1}{2}-\mu-\kappa)} > 0</math> || <syntaxhighlight lang=mathematica>WhittakerW(kappa, mu, z) = (exp(-(1)/(2)*z)*(z)^(kappa + 1))/(GAMMA((1)/(2)+ mu - kappa)*GAMMA((1)/(2)- mu - kappa))* int((WhittakerW(- kappa, mu, t)*exp(-(1)/(2)*t)*(t)^(- kappa - 1))/(t + z), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerW[\[Kappa], \[Mu], z] == Divide[Exp[-Divide[1,2]*z]*(z)^(\[Kappa]+ 1),Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]*Gamma[Divide[1,2]- \[Mu]- \[Kappa]]]* Integrate[Divide[WhittakerW[- \[Kappa], \[Mu], t]*Exp[-Divide[1,2]*t]*(t)^(- \[Kappa]- 1),t + z], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Manual Skip! || Successful [Tested: 300] | ||
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| [https://dlmf.nist.gov/13.16.E7 13.16.E7] | | | [https://dlmf.nist.gov/13.16.E7 13.16.E7] || <math qid="Q4558">\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{(-1)^{n}e^{-\frac{1}{2}z}z^{\frac{1}{2}-\mu-n}}{\EulerGamma@{1+2\mu}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}\*\int_{0}^{\infty}\frac{\WhittakerconfhyperM{-\kappa}{\mu}@{t}e^{-\frac{1}{2}t}t^{n+\mu-\frac{1}{2}}}{t+z}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{(-1)^{n}e^{-\frac{1}{2}z}z^{\frac{1}{2}-\mu-n}}{\EulerGamma@{1+2\mu}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}\*\int_{0}^{\infty}\frac{\WhittakerconfhyperM{-\kappa}{\mu}@{t}e^{-\frac{1}{2}t}t^{n+\mu-\frac{1}{2}}}{t+z}\diff{t}</syntaxhighlight> || <math>|\phase@@{z}| < \pi, -\realpart@{1+2\mu} < n, n < \abs{\realpart@@{\mu}}+\realpart@@{\kappa}, \abs{\realpart@@{\mu}}+\realpart@@{\kappa} < \tfrac{1}{2}, \realpart@@{(1+2\mu)} > 0, \realpart@@{(\frac{1}{2}-\mu-\kappa)} > 0</math> || <syntaxhighlight lang=mathematica>WhittakerW(kappa, mu, z) = ((- 1)^(n)* exp(-(1)/(2)*z)*(z)^((1)/(2)- mu - n))/(GAMMA(1 + 2*mu)*GAMMA((1)/(2)- mu - kappa))* int((WhittakerM(- kappa, mu, t)*exp(-(1)/(2)*t)*(t)^(n + mu -(1)/(2)))/(t + z), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerW[\[Kappa], \[Mu], z] == Divide[(- 1)^(n)* Exp[-Divide[1,2]*z]*(z)^(Divide[1,2]- \[Mu]- n),Gamma[1 + 2*\[Mu]]*Gamma[Divide[1,2]- \[Mu]- \[Kappa]]]* Integrate[Divide[WhittakerM[- \[Kappa], \[Mu], t]*Exp[-Divide[1,2]*t]*(t)^(n + \[Mu]-Divide[1,2]),t + z], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Manual Skip! || Skipped - Because timed out | ||
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| [https://dlmf.nist.gov/13.16.E8 13.16.E8] | | | [https://dlmf.nist.gov/13.16.E8 13.16.E8] || <math qid="Q4559">\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{2\sqrt{z}e^{-\frac{1}{2}z}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}\*\int_{0}^{\infty}e^{-t}t^{-\kappa-\frac{1}{2}}\modBesselK{2\mu}@{2\sqrt{zt}}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{2\sqrt{z}e^{-\frac{1}{2}z}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}\*\int_{0}^{\infty}e^{-t}t^{-\kappa-\frac{1}{2}}\modBesselK{2\mu}@{2\sqrt{zt}}\diff{t}</syntaxhighlight> || <math>\realpart@{\mu-\kappa}+\tfrac{1}{2} > 0, \realpart@@{(\frac{1}{2}+\mu-\kappa)} > 0, \realpart@@{(\frac{1}{2}-\mu-\kappa)} > 0</math> || <syntaxhighlight lang=mathematica>WhittakerW(kappa, mu, z) = (2*sqrt(z)*exp(-(1)/(2)*z))/(GAMMA((1)/(2)+ mu - kappa)*GAMMA((1)/(2)- mu - kappa))* int(exp(- t)*(t)^(- kappa -(1)/(2))* BesselK(2*mu, 2*sqrt(z*t)), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerW[\[Kappa], \[Mu], z] == Divide[2*Sqrt[z]*Exp[-Divide[1,2]*z],Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]*Gamma[Divide[1,2]- \[Mu]- \[Kappa]]]* Integrate[Exp[- t]*(t)^(- \[Kappa]-Divide[1,2])* BesselK[2*\[Mu], 2*Sqrt[z*t]], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Aborted || - || Successful [Tested: 252] | ||
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| [https://dlmf.nist.gov/13.16.E9 13.16.E9] | | | [https://dlmf.nist.gov/13.16.E9 13.16.E9] || <math qid="Q4560">\WhittakerconfhyperW{\kappa}{\mu}@{z} = e^{-\frac{1}{2}z}z^{\kappa+c}\*\int_{0}^{\infty}e^{-zt}t^{c-1}\genhyperOlverF{2}{1}@@{\tfrac{1}{2}+\mu-\kappa,\tfrac{1}{2}-\mu-\kappa}{c}{-t}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperW{\kappa}{\mu}@{z} = e^{-\frac{1}{2}z}z^{\kappa+c}\*\int_{0}^{\infty}e^{-zt}t^{c-1}\genhyperOlverF{2}{1}@@{\tfrac{1}{2}+\mu-\kappa,\tfrac{1}{2}-\mu-\kappa}{c}{-t}\diff{t}</syntaxhighlight> || <math>|\phase{z}| < \frac{1}{2}\pi</math> || <syntaxhighlight lang=mathematica>WhittakerW(kappa, mu, z) = exp(-(1)/(2)*z)*(z)^(kappa + c)* int(exp(- z*t)*(t)^(c - 1)* hypergeom([(1)/(2)+ mu - kappa ,(1)/(2)- mu - kappa], [c], - t), t = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerW[\[Kappa], \[Mu], z] == Exp[-Divide[1,2]*z]*(z)^(\[Kappa]+ c)* Integrate[Exp[- z*t]*(t)^(c - 1)* HypergeometricPFQRegularized[{Divide[1,2]+ \[Mu]- \[Kappa],Divide[1,2]- \[Mu]- \[Kappa]}, {c}, - t], {t, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Manual Skip! || Skipped - Because timed out | ||
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| [https://dlmf.nist.gov/13.16.E10 13.16.E10] | | | [https://dlmf.nist.gov/13.16.E10 13.16.E10] || <math qid="Q4561">\frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{e^{+\pi\iunit}z} = \frac{e^{\frac{1}{2}z+(\frac{1}{2}+\mu)\pi\iunit}}{2\pi\iunit\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{t-\kappa}\EulerGamma@{\frac{1}{2}+\mu-t}}{\EulerGamma@{\frac{1}{2}+\mu+t}}z^{t}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{e^{+\pi\iunit}z} = \frac{e^{\frac{1}{2}z+(\frac{1}{2}+\mu)\pi\iunit}}{2\pi\iunit\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{t-\kappa}\EulerGamma@{\frac{1}{2}+\mu-t}}{\EulerGamma@{\frac{1}{2}+\mu+t}}z^{t}\diff{t}</syntaxhighlight> || <math>|\phase{z}| < \tfrac{1}{2}\pi, \realpart@@{(1+2\mu)} > 0, \realpart@@{(\frac{1}{2}+\mu-\kappa)} > 0, \realpart@@{(t-\kappa)} > 0, \realpart@@{(\frac{1}{2}+\mu-t)} > 0, \realpart@@{(\frac{1}{2}+\mu+t)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(GAMMA(1 + 2*mu))*WhittakerM(kappa, mu, exp(+ Pi*I)*z) = (exp((1)/(2)*z +((1)/(2)+ mu)*Pi*I))/(2*Pi*I*GAMMA((1)/(2)+ mu - kappa))* int((GAMMA(t - kappa)*GAMMA((1)/(2)+ mu - t))/(GAMMA((1)/(2)+ mu + t))*(z)^(t), t = - I*infinity..I*infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,Gamma[1 + 2*\[Mu]]]*WhittakerM[\[Kappa], \[Mu], Exp[+ Pi*I]*z] == Divide[Exp[Divide[1,2]*z +(Divide[1,2]+ \[Mu])*Pi*I],2*Pi*I*Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]]* Integrate[Divide[Gamma[t - \[Kappa]]*Gamma[Divide[1,2]+ \[Mu]- t],Gamma[Divide[1,2]+ \[Mu]+ t]]*(z)^(t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out | ||
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| [https://dlmf.nist.gov/13.16.E10 13.16.E10] | | | [https://dlmf.nist.gov/13.16.E10 13.16.E10] || <math qid="Q4561">\frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{e^{-\pi\iunit}z} = \frac{e^{\frac{1}{2}z-(\frac{1}{2}+\mu)\pi\iunit}}{2\pi\iunit\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{t-\kappa}\EulerGamma@{\frac{1}{2}+\mu-t}}{\EulerGamma@{\frac{1}{2}+\mu+t}}z^{t}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{e^{-\pi\iunit}z} = \frac{e^{\frac{1}{2}z-(\frac{1}{2}+\mu)\pi\iunit}}{2\pi\iunit\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{t-\kappa}\EulerGamma@{\frac{1}{2}+\mu-t}}{\EulerGamma@{\frac{1}{2}+\mu+t}}z^{t}\diff{t}</syntaxhighlight> || <math>|\phase{z}| < \tfrac{1}{2}\pi, \realpart@@{(1+2\mu)} > 0, \realpart@@{(\frac{1}{2}+\mu-\kappa)} > 0, \realpart@@{(t-\kappa)} > 0, \realpart@@{(\frac{1}{2}+\mu-t)} > 0, \realpart@@{(\frac{1}{2}+\mu+t)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(GAMMA(1 + 2*mu))*WhittakerM(kappa, mu, exp(- Pi*I)*z) = (exp((1)/(2)*z -((1)/(2)+ mu)*Pi*I))/(2*Pi*I*GAMMA((1)/(2)+ mu - kappa))* int((GAMMA(t - kappa)*GAMMA((1)/(2)+ mu - t))/(GAMMA((1)/(2)+ mu + t))*(z)^(t), t = - I*infinity..I*infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,Gamma[1 + 2*\[Mu]]]*WhittakerM[\[Kappa], \[Mu], Exp[- Pi*I]*z] == Divide[Exp[Divide[1,2]*z -(Divide[1,2]+ \[Mu])*Pi*I],2*Pi*I*Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]]* Integrate[Divide[Gamma[t - \[Kappa]]*Gamma[Divide[1,2]+ \[Mu]- t],Gamma[Divide[1,2]+ \[Mu]+ t]]*(z)^(t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out | ||
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| [https://dlmf.nist.gov/13.16.E11 13.16.E11] | | | [https://dlmf.nist.gov/13.16.E11 13.16.E11] || <math qid="Q4562">\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{e^{-\frac{1}{2}z}}{2\pi\iunit}\*\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{\frac{1}{2}+\mu+t}\EulerGamma@{\frac{1}{2}-\mu+t}\EulerGamma@{-\kappa-t}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}z^{-t}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{e^{-\frac{1}{2}z}}{2\pi\iunit}\*\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{\frac{1}{2}+\mu+t}\EulerGamma@{\frac{1}{2}-\mu+t}\EulerGamma@{-\kappa-t}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}z^{-t}\diff{t}</syntaxhighlight> || <math>|\phase{z}| < \tfrac{3}{2}\pi, \realpart@@{(\frac{1}{2}+\mu+t)} > 0, \realpart@@{(\frac{1}{2}-\mu+t)} > 0, \realpart@@{(-\kappa-t)} > 0, \realpart@@{(\frac{1}{2}+\mu-\kappa)} > 0, \realpart@@{(\frac{1}{2}-\mu-\kappa)} > 0</math> || <syntaxhighlight lang=mathematica>WhittakerW(kappa, mu, z) = (exp(-(1)/(2)*z))/(2*Pi*I)* int((GAMMA((1)/(2)+ mu + t)*GAMMA((1)/(2)- mu + t)*GAMMA(- kappa - t))/(GAMMA((1)/(2)+ mu - kappa)*GAMMA((1)/(2)- mu - kappa))*(z)^(- t), t = - I*infinity..I*infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerW[\[Kappa], \[Mu], z] == Divide[Exp[-Divide[1,2]*z],2*Pi*I]* Integrate[Divide[Gamma[Divide[1,2]+ \[Mu]+ t]*Gamma[Divide[1,2]- \[Mu]+ t]*Gamma[- \[Kappa]- t],Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]*Gamma[Divide[1,2]- \[Mu]- \[Kappa]]]*(z)^(- t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out | ||
|- | |- | ||
| [https://dlmf.nist.gov/13.16.E12 13.16.E12] | | | [https://dlmf.nist.gov/13.16.E12 13.16.E12] || <math qid="Q4563">\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{e^{\frac{1}{2}z}}{2\pi\iunit}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{\frac{1}{2}+\mu+t}\EulerGamma@{\frac{1}{2}-\mu+t}}{\EulerGamma@{1-\kappa+t}}z^{-t}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{e^{\frac{1}{2}z}}{2\pi\iunit}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{\frac{1}{2}+\mu+t}\EulerGamma@{\frac{1}{2}-\mu+t}}{\EulerGamma@{1-\kappa+t}}z^{-t}\diff{t}</syntaxhighlight> || <math>|\phase{z}| < \tfrac{1}{2}\pi, \realpart@@{(\frac{1}{2}+\mu+t)} > 0, \realpart@@{(\frac{1}{2}-\mu+t)} > 0, \realpart@@{(1-\kappa+t)} > 0</math> || <syntaxhighlight lang=mathematica>WhittakerW(kappa, mu, z) = (exp((1)/(2)*z))/(2*Pi*I)*int((GAMMA((1)/(2)+ mu + t)*GAMMA((1)/(2)- mu + t))/(GAMMA(1 - kappa + t))*(z)^(- t), t = - I*infinity..I*infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerW[\[Kappa], \[Mu], z] == Divide[Exp[Divide[1,2]*z],2*Pi*I]*Integrate[Divide[Gamma[Divide[1,2]+ \[Mu]+ t]*Gamma[Divide[1,2]- \[Mu]+ t],Gamma[1 - \[Kappa]+ t]]*(z)^(- t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out | ||
|} | |} | ||
</div> | </div> | ||
Latest revision as of 11:34, 28 June 2021
| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 13.16.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \WhittakerconfhyperM{\kappa}{\mu}@{z} = \frac{\EulerGamma@{1+2\mu}z^{\mu+\frac{1}{2}}2^{-2\mu}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}+\mu+\kappa}}\*\int_{-1}^{1}e^{\frac{1}{2}zt}(1+t)^{\mu-\frac{1}{2}-\kappa}(1-t)^{\mu-\frac{1}{2}+\kappa}\diff{t}}
\WhittakerconfhyperM{\kappa}{\mu}@{z} = \frac{\EulerGamma@{1+2\mu}z^{\mu+\frac{1}{2}}2^{-2\mu}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}+\mu+\kappa}}\*\int_{-1}^{1}e^{\frac{1}{2}zt}(1+t)^{\mu-\frac{1}{2}-\kappa}(1-t)^{\mu-\frac{1}{2}+\kappa}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{\mu}+\tfrac{1}{2} > \left|\realpart@@{\kappa}\right|, \realpart@@{(1+2\mu)} > 0, \realpart@@{(\frac{1}{2}+\mu-\kappa)} > 0, \realpart@@{(\frac{1}{2}+\mu+\kappa)} > 0} | WhittakerM(kappa, mu, z) = (GAMMA(1 + 2*mu)*(z)^(mu +(1)/(2))* (2)^(- 2*mu))/(GAMMA((1)/(2)+ mu - kappa)*GAMMA((1)/(2)+ mu + kappa))* int(exp((1)/(2)*z*t)*(1 + t)^(mu -(1)/(2)- kappa)*(1 - t)^(mu -(1)/(2)+ kappa), t = - 1..1)
|
WhittakerM[\[Kappa], \[Mu], z] == Divide[Gamma[1 + 2*\[Mu]]*(z)^(\[Mu]+Divide[1,2])* (2)^(- 2*\[Mu]),Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]*Gamma[Divide[1,2]+ \[Mu]+ \[Kappa]]]* Integrate[Exp[Divide[1,2]*z*t]*(1 + t)^(\[Mu]-Divide[1,2]- \[Kappa])*(1 - t)^(\[Mu]-Divide[1,2]+ \[Kappa]), {t, - 1, 1}, GenerateConditions->None]
|
Failure | Successful | Skipped - Because timed out | Successful [Tested: 252] |
| 13.16.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \WhittakerconfhyperM{\kappa}{\mu}@{z} = \frac{\EulerGamma@{1+2\mu}z^{\lambda}}{\EulerGamma@{1+2\mu-2\lambda}\EulerGamma@{2\lambda}}\*\int_{0}^{1}\WhittakerconfhyperM{\kappa-\lambda}{\mu-\lambda}@{zt}e^{\frac{1}{2}z(t-1)}t^{\mu-\lambda-\frac{1}{2}}{(1-t)^{2\lambda-1}}\diff{t}}
\WhittakerconfhyperM{\kappa}{\mu}@{z} = \frac{\EulerGamma@{1+2\mu}z^{\lambda}}{\EulerGamma@{1+2\mu-2\lambda}\EulerGamma@{2\lambda}}\*\int_{0}^{1}\WhittakerconfhyperM{\kappa-\lambda}{\mu-\lambda}@{zt}e^{\frac{1}{2}z(t-1)}t^{\mu-\lambda-\frac{1}{2}}{(1-t)^{2\lambda-1}}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{\mu}+\tfrac{1}{2} > \realpart@@{\lambda}, \realpart@@{\lambda} > 0, \realpart@@{(1+2\mu)} > 0, \realpart@@{(1+2\mu-2\lambda)} > 0, \realpart@@{(2\lambda)} > 0} | WhittakerM(kappa, mu, z) = (GAMMA(1 + 2*mu)*(z)^(lambda))/(GAMMA(1 + 2*mu - 2*lambda)*GAMMA(2*lambda))* int(WhittakerM(kappa - lambda, mu - lambda, z*t)*exp((1)/(2)*z*(t - 1))*(t)^(mu - lambda -(1)/(2))*(1 - t)^(2*lambda - 1), t = 0..1)
|
WhittakerM[\[Kappa], \[Mu], z] == Divide[Gamma[1 + 2*\[Mu]]*(z)^\[Lambda],Gamma[1 + 2*\[Mu]- 2*\[Lambda]]*Gamma[2*\[Lambda]]]* Integrate[WhittakerM[\[Kappa]- \[Lambda], \[Mu]- \[Lambda], z*t]*Exp[Divide[1,2]*z*(t - 1)]*(t)^(\[Mu]- \[Lambda]-Divide[1,2])*(1 - t)^(2*\[Lambda]- 1), {t, 0, 1}, GenerateConditions->None]
|
Failure | Aborted | Skipped - Because timed out | Skipped - Because timed out |
| 13.16.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{z} = \frac{\sqrt{z}e^{\frac{1}{2}z}}{\EulerGamma@{\frac{1}{2}+\mu+\kappa}}\int_{0}^{\infty}e^{-t}t^{\kappa-\frac{1}{2}}\BesselJ{2\mu}@{2\sqrt{zt}}\diff{t}}
\frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{z} = \frac{\sqrt{z}e^{\frac{1}{2}z}}{\EulerGamma@{\frac{1}{2}+\mu+\kappa}}\int_{0}^{\infty}e^{-t}t^{\kappa-\frac{1}{2}}\BesselJ{2\mu}@{2\sqrt{zt}}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@{\kappa+\mu}+\tfrac{1}{2} > 0, \realpart@@{((2\mu)+k+1)} > 0, \realpart@@{(1+2\mu)} > 0, \realpart@@{(\frac{1}{2}+\mu+\kappa)} > 0} | (1)/(GAMMA(1 + 2*mu))*WhittakerM(kappa, mu, z) = (sqrt(z)*exp((1)/(2)*z))/(GAMMA((1)/(2)+ mu + kappa))*int(exp(- t)*(t)^(kappa -(1)/(2))* BesselJ(2*mu, 2*sqrt(z*t)), t = 0..infinity)
|
Divide[1,Gamma[1 + 2*\[Mu]]]*WhittakerM[\[Kappa], \[Mu], z] == Divide[Sqrt[z]*Exp[Divide[1,2]*z],Gamma[Divide[1,2]+ \[Mu]+ \[Kappa]]]*Integrate[Exp[- t]*(t)^(\[Kappa]-Divide[1,2])* BesselJ[2*\[Mu], 2*Sqrt[z*t]], {t, 0, Infinity}, GenerateConditions->None]
|
Successful | Aborted | - | Skipped - Because timed out |
| 13.16.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{z} = \frac{\sqrt{z}e^{-\frac{1}{2}z}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*\int_{0}^{\infty}e^{-t}t^{-\kappa-\frac{1}{2}}\modBesselI{2\mu}@{2\sqrt{zt}}\diff{t}}
\frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{z} = \frac{\sqrt{z}e^{-\frac{1}{2}z}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*\int_{0}^{\infty}e^{-t}t^{-\kappa-\frac{1}{2}}\modBesselI{2\mu}@{2\sqrt{zt}}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\kappa-\mu)-\tfrac{1}{2}} < 0, \realpart@@{(1+2\mu)} > 0, \realpart@@{(\frac{1}{2}+\mu-\kappa)} > 0, \realpart@@{((2\mu)+k+1)} > 0} | (1)/(GAMMA(1 + 2*mu))*WhittakerM(kappa, mu, z) = (sqrt(z)*exp(-(1)/(2)*z))/(GAMMA((1)/(2)+ mu - kappa))* int(exp(- t)*(t)^(- kappa -(1)/(2))* BesselI(2*mu, 2*sqrt(z*t)), t = 0..infinity)
|
Divide[1,Gamma[1 + 2*\[Mu]]]*WhittakerM[\[Kappa], \[Mu], z] == Divide[Sqrt[z]*Exp[-Divide[1,2]*z],Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]]* Integrate[Exp[- t]*(t)^(- \[Kappa]-Divide[1,2])* BesselI[2*\[Mu], 2*Sqrt[z*t]], {t, 0, Infinity}, GenerateConditions->None]
|
Failure | Successful | Failed [42 / 300] Result: .5483729950e-2+.5411197480e-1*I
Test Values: {kappa = -3/2, mu = 2, z = 1/2*3^(1/2)+1/2*I}
Result: .2482822497e-1-.2550894001e-1*I
Test Values: {kappa = -3/2, mu = 2, z = -1/2+1/2*I*3^(1/2)}
... skip entries to safe data |
Successful [Tested: 300] |
| 13.16.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{z^{\mu+\frac{1}{2}}2^{-2\mu}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*\int_{1}^{\infty}e^{-\frac{1}{2}zt}(t-1)^{\mu-\frac{1}{2}-\kappa}(t+1)^{\mu-\frac{1}{2}+\kappa}\diff{t}}
\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{z^{\mu+\frac{1}{2}}2^{-2\mu}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*\int_{1}^{\infty}e^{-\frac{1}{2}zt}(t-1)^{\mu-\frac{1}{2}-\kappa}(t+1)^{\mu-\frac{1}{2}+\kappa}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{\mu}+\tfrac{1}{2} > \realpart@@{\kappa}, |\phase{z}| < \frac{1}{2}\pi, \realpart@@{(\frac{1}{2}+\mu-\kappa)} > 0} | WhittakerW(kappa, mu, z) = ((z)^(mu +(1)/(2))* (2)^(- 2*mu))/(GAMMA((1)/(2)+ mu - kappa))* int(exp(-(1)/(2)*z*t)*(t - 1)^(mu -(1)/(2)- kappa)*(t + 1)^(mu -(1)/(2)+ kappa), t = 1..infinity)
|
WhittakerW[\[Kappa], \[Mu], z] == Divide[(z)^(\[Mu]+Divide[1,2])* (2)^(- 2*\[Mu]),Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]]* Integrate[Exp[-Divide[1,2]*z*t]*(t - 1)^(\[Mu]-Divide[1,2]- \[Kappa])*(t + 1)^(\[Mu]-Divide[1,2]+ \[Kappa]), {t, 1, Infinity}, GenerateConditions->None]
|
Failure | Aborted | Skipped - Because timed out | Successful [Tested: 300] |
| 13.16.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{e^{-\frac{1}{2}z}z^{\kappa+1}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}\*\int_{0}^{\infty}\frac{\WhittakerconfhyperW{-\kappa}{\mu}@{t}e^{-\frac{1}{2}t}t^{-\kappa-1}}{t+z}\diff{t}}
\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{e^{-\frac{1}{2}z}z^{\kappa+1}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}\*\int_{0}^{\infty}\frac{\WhittakerconfhyperW{-\kappa}{\mu}@{t}e^{-\frac{1}{2}t}t^{-\kappa-1}}{t+z}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\phase{z}| < \pi, \realpart@{\frac{1}{2}+\mu-\kappa} > \max\left(2\realpart@@{\mu}, \realpart@@{(\frac{1}{2}+\mu-\kappa)} > 0, \realpart@@{(\frac{1}{2}-\mu-\kappa)} > 0} | WhittakerW(kappa, mu, z) = (exp(-(1)/(2)*z)*(z)^(kappa + 1))/(GAMMA((1)/(2)+ mu - kappa)*GAMMA((1)/(2)- mu - kappa))* int((WhittakerW(- kappa, mu, t)*exp(-(1)/(2)*t)*(t)^(- kappa - 1))/(t + z), t = 0..infinity)
|
WhittakerW[\[Kappa], \[Mu], z] == Divide[Exp[-Divide[1,2]*z]*(z)^(\[Kappa]+ 1),Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]*Gamma[Divide[1,2]- \[Mu]- \[Kappa]]]* Integrate[Divide[WhittakerW[- \[Kappa], \[Mu], t]*Exp[-Divide[1,2]*t]*(t)^(- \[Kappa]- 1),t + z], {t, 0, Infinity}, GenerateConditions->None]
|
Failure | Aborted | Manual Skip! | Successful [Tested: 300] |
| 13.16.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{(-1)^{n}e^{-\frac{1}{2}z}z^{\frac{1}{2}-\mu-n}}{\EulerGamma@{1+2\mu}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}\*\int_{0}^{\infty}\frac{\WhittakerconfhyperM{-\kappa}{\mu}@{t}e^{-\frac{1}{2}t}t^{n+\mu-\frac{1}{2}}}{t+z}\diff{t}}
\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{(-1)^{n}e^{-\frac{1}{2}z}z^{\frac{1}{2}-\mu-n}}{\EulerGamma@{1+2\mu}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}\*\int_{0}^{\infty}\frac{\WhittakerconfhyperM{-\kappa}{\mu}@{t}e^{-\frac{1}{2}t}t^{n+\mu-\frac{1}{2}}}{t+z}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\phase@@{z}| < \pi, -\realpart@{1+2\mu} < n, n < \abs{\realpart@@{\mu}}+\realpart@@{\kappa}, \abs{\realpart@@{\mu}}+\realpart@@{\kappa} < \tfrac{1}{2}, \realpart@@{(1+2\mu)} > 0, \realpart@@{(\frac{1}{2}-\mu-\kappa)} > 0} | WhittakerW(kappa, mu, z) = ((- 1)^(n)* exp(-(1)/(2)*z)*(z)^((1)/(2)- mu - n))/(GAMMA(1 + 2*mu)*GAMMA((1)/(2)- mu - kappa))* int((WhittakerM(- kappa, mu, t)*exp(-(1)/(2)*t)*(t)^(n + mu -(1)/(2)))/(t + z), t = 0..infinity)
|
WhittakerW[\[Kappa], \[Mu], z] == Divide[(- 1)^(n)* Exp[-Divide[1,2]*z]*(z)^(Divide[1,2]- \[Mu]- n),Gamma[1 + 2*\[Mu]]*Gamma[Divide[1,2]- \[Mu]- \[Kappa]]]* Integrate[Divide[WhittakerM[- \[Kappa], \[Mu], t]*Exp[-Divide[1,2]*t]*(t)^(n + \[Mu]-Divide[1,2]),t + z], {t, 0, Infinity}, GenerateConditions->None]
|
Failure | Aborted | Manual Skip! | Skipped - Because timed out |
| 13.16.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{2\sqrt{z}e^{-\frac{1}{2}z}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}\*\int_{0}^{\infty}e^{-t}t^{-\kappa-\frac{1}{2}}\modBesselK{2\mu}@{2\sqrt{zt}}\diff{t}}
\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{2\sqrt{z}e^{-\frac{1}{2}z}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}\*\int_{0}^{\infty}e^{-t}t^{-\kappa-\frac{1}{2}}\modBesselK{2\mu}@{2\sqrt{zt}}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@{\mu-\kappa}+\tfrac{1}{2} > 0, \realpart@@{(\frac{1}{2}+\mu-\kappa)} > 0, \realpart@@{(\frac{1}{2}-\mu-\kappa)} > 0} | WhittakerW(kappa, mu, z) = (2*sqrt(z)*exp(-(1)/(2)*z))/(GAMMA((1)/(2)+ mu - kappa)*GAMMA((1)/(2)- mu - kappa))* int(exp(- t)*(t)^(- kappa -(1)/(2))* BesselK(2*mu, 2*sqrt(z*t)), t = 0..infinity)
|
WhittakerW[\[Kappa], \[Mu], z] == Divide[2*Sqrt[z]*Exp[-Divide[1,2]*z],Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]*Gamma[Divide[1,2]- \[Mu]- \[Kappa]]]* Integrate[Exp[- t]*(t)^(- \[Kappa]-Divide[1,2])* BesselK[2*\[Mu], 2*Sqrt[z*t]], {t, 0, Infinity}, GenerateConditions->None]
|
Successful | Aborted | - | Successful [Tested: 252] |
| 13.16.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \WhittakerconfhyperW{\kappa}{\mu}@{z} = e^{-\frac{1}{2}z}z^{\kappa+c}\*\int_{0}^{\infty}e^{-zt}t^{c-1}\genhyperOlverF{2}{1}@@{\tfrac{1}{2}+\mu-\kappa,\tfrac{1}{2}-\mu-\kappa}{c}{-t}\diff{t}}
\WhittakerconfhyperW{\kappa}{\mu}@{z} = e^{-\frac{1}{2}z}z^{\kappa+c}\*\int_{0}^{\infty}e^{-zt}t^{c-1}\genhyperOlverF{2}{1}@@{\tfrac{1}{2}+\mu-\kappa,\tfrac{1}{2}-\mu-\kappa}{c}{-t}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\phase{z}| < \frac{1}{2}\pi} | WhittakerW(kappa, mu, z) = exp(-(1)/(2)*z)*(z)^(kappa + c)* int(exp(- z*t)*(t)^(c - 1)* hypergeom([(1)/(2)+ mu - kappa ,(1)/(2)- mu - kappa], [c], - t), t = 0..infinity)
|
WhittakerW[\[Kappa], \[Mu], z] == Exp[-Divide[1,2]*z]*(z)^(\[Kappa]+ c)* Integrate[Exp[- z*t]*(t)^(c - 1)* HypergeometricPFQRegularized[{Divide[1,2]+ \[Mu]- \[Kappa],Divide[1,2]- \[Mu]- \[Kappa]}, {c}, - t], {t, 0, Infinity}, GenerateConditions->None]
|
Failure | Aborted | Manual Skip! | Skipped - Because timed out |
| 13.16.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{e^{+\pi\iunit}z} = \frac{e^{\frac{1}{2}z+(\frac{1}{2}+\mu)\pi\iunit}}{2\pi\iunit\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{t-\kappa}\EulerGamma@{\frac{1}{2}+\mu-t}}{\EulerGamma@{\frac{1}{2}+\mu+t}}z^{t}\diff{t}}
\frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{e^{+\pi\iunit}z} = \frac{e^{\frac{1}{2}z+(\frac{1}{2}+\mu)\pi\iunit}}{2\pi\iunit\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{t-\kappa}\EulerGamma@{\frac{1}{2}+\mu-t}}{\EulerGamma@{\frac{1}{2}+\mu+t}}z^{t}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\phase{z}| < \tfrac{1}{2}\pi, \realpart@@{(1+2\mu)} > 0, \realpart@@{(\frac{1}{2}+\mu-\kappa)} > 0, \realpart@@{(t-\kappa)} > 0, \realpart@@{(\frac{1}{2}+\mu-t)} > 0, \realpart@@{(\frac{1}{2}+\mu+t)} > 0} | (1)/(GAMMA(1 + 2*mu))*WhittakerM(kappa, mu, exp(+ Pi*I)*z) = (exp((1)/(2)*z +((1)/(2)+ mu)*Pi*I))/(2*Pi*I*GAMMA((1)/(2)+ mu - kappa))* int((GAMMA(t - kappa)*GAMMA((1)/(2)+ mu - t))/(GAMMA((1)/(2)+ mu + t))*(z)^(t), t = - I*infinity..I*infinity)
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Divide[1,Gamma[1 + 2*\[Mu]]]*WhittakerM[\[Kappa], \[Mu], Exp[+ Pi*I]*z] == Divide[Exp[Divide[1,2]*z +(Divide[1,2]+ \[Mu])*Pi*I],2*Pi*I*Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]]* Integrate[Divide[Gamma[t - \[Kappa]]*Gamma[Divide[1,2]+ \[Mu]- t],Gamma[Divide[1,2]+ \[Mu]+ t]]*(z)^(t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]
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Failure | Aborted | Skipped - Because timed out | Skipped - Because timed out |
| 13.16.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{e^{-\pi\iunit}z} = \frac{e^{\frac{1}{2}z-(\frac{1}{2}+\mu)\pi\iunit}}{2\pi\iunit\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{t-\kappa}\EulerGamma@{\frac{1}{2}+\mu-t}}{\EulerGamma@{\frac{1}{2}+\mu+t}}z^{t}\diff{t}}
\frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{e^{-\pi\iunit}z} = \frac{e^{\frac{1}{2}z-(\frac{1}{2}+\mu)\pi\iunit}}{2\pi\iunit\EulerGamma@{\frac{1}{2}+\mu-\kappa}}\*\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{t-\kappa}\EulerGamma@{\frac{1}{2}+\mu-t}}{\EulerGamma@{\frac{1}{2}+\mu+t}}z^{t}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\phase{z}| < \tfrac{1}{2}\pi, \realpart@@{(1+2\mu)} > 0, \realpart@@{(\frac{1}{2}+\mu-\kappa)} > 0, \realpart@@{(t-\kappa)} > 0, \realpart@@{(\frac{1}{2}+\mu-t)} > 0, \realpart@@{(\frac{1}{2}+\mu+t)} > 0} | (1)/(GAMMA(1 + 2*mu))*WhittakerM(kappa, mu, exp(- Pi*I)*z) = (exp((1)/(2)*z -((1)/(2)+ mu)*Pi*I))/(2*Pi*I*GAMMA((1)/(2)+ mu - kappa))* int((GAMMA(t - kappa)*GAMMA((1)/(2)+ mu - t))/(GAMMA((1)/(2)+ mu + t))*(z)^(t), t = - I*infinity..I*infinity)
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Divide[1,Gamma[1 + 2*\[Mu]]]*WhittakerM[\[Kappa], \[Mu], Exp[- Pi*I]*z] == Divide[Exp[Divide[1,2]*z -(Divide[1,2]+ \[Mu])*Pi*I],2*Pi*I*Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]]* Integrate[Divide[Gamma[t - \[Kappa]]*Gamma[Divide[1,2]+ \[Mu]- t],Gamma[Divide[1,2]+ \[Mu]+ t]]*(z)^(t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]
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Failure | Aborted | Skipped - Because timed out | Skipped - Because timed out |
| 13.16.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{e^{-\frac{1}{2}z}}{2\pi\iunit}\*\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{\frac{1}{2}+\mu+t}\EulerGamma@{\frac{1}{2}-\mu+t}\EulerGamma@{-\kappa-t}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}z^{-t}\diff{t}}
\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{e^{-\frac{1}{2}z}}{2\pi\iunit}\*\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{\frac{1}{2}+\mu+t}\EulerGamma@{\frac{1}{2}-\mu+t}\EulerGamma@{-\kappa-t}}{\EulerGamma@{\frac{1}{2}+\mu-\kappa}\EulerGamma@{\frac{1}{2}-\mu-\kappa}}z^{-t}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\phase{z}| < \tfrac{3}{2}\pi, \realpart@@{(\frac{1}{2}+\mu+t)} > 0, \realpart@@{(\frac{1}{2}-\mu+t)} > 0, \realpart@@{(-\kappa-t)} > 0, \realpart@@{(\frac{1}{2}+\mu-\kappa)} > 0, \realpart@@{(\frac{1}{2}-\mu-\kappa)} > 0} | WhittakerW(kappa, mu, z) = (exp(-(1)/(2)*z))/(2*Pi*I)* int((GAMMA((1)/(2)+ mu + t)*GAMMA((1)/(2)- mu + t)*GAMMA(- kappa - t))/(GAMMA((1)/(2)+ mu - kappa)*GAMMA((1)/(2)- mu - kappa))*(z)^(- t), t = - I*infinity..I*infinity)
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WhittakerW[\[Kappa], \[Mu], z] == Divide[Exp[-Divide[1,2]*z],2*Pi*I]* Integrate[Divide[Gamma[Divide[1,2]+ \[Mu]+ t]*Gamma[Divide[1,2]- \[Mu]+ t]*Gamma[- \[Kappa]- t],Gamma[Divide[1,2]+ \[Mu]- \[Kappa]]*Gamma[Divide[1,2]- \[Mu]- \[Kappa]]]*(z)^(- t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]
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Failure | Aborted | Skipped - Because timed out | Skipped - Because timed out |
| 13.16.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{e^{\frac{1}{2}z}}{2\pi\iunit}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{\frac{1}{2}+\mu+t}\EulerGamma@{\frac{1}{2}-\mu+t}}{\EulerGamma@{1-\kappa+t}}z^{-t}\diff{t}}
\WhittakerconfhyperW{\kappa}{\mu}@{z} = \frac{e^{\frac{1}{2}z}}{2\pi\iunit}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{\frac{1}{2}+\mu+t}\EulerGamma@{\frac{1}{2}-\mu+t}}{\EulerGamma@{1-\kappa+t}}z^{-t}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\phase{z}| < \tfrac{1}{2}\pi, \realpart@@{(\frac{1}{2}+\mu+t)} > 0, \realpart@@{(\frac{1}{2}-\mu+t)} > 0, \realpart@@{(1-\kappa+t)} > 0} | WhittakerW(kappa, mu, z) = (exp((1)/(2)*z))/(2*Pi*I)*int((GAMMA((1)/(2)+ mu + t)*GAMMA((1)/(2)- mu + t))/(GAMMA(1 - kappa + t))*(z)^(- t), t = - I*infinity..I*infinity)
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WhittakerW[\[Kappa], \[Mu], z] == Divide[Exp[Divide[1,2]*z],2*Pi*I]*Integrate[Divide[Gamma[Divide[1,2]+ \[Mu]+ t]*Gamma[Divide[1,2]- \[Mu]+ t],Gamma[1 - \[Kappa]+ t]]*(z)^(- t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]
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Failure | Aborted | Skipped - Because timed out | Skipped - Because timed out |