13.18: Difference between revisions
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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/13.18.E1 13.18.E1] | | | [https://dlmf.nist.gov/13.18.E1 13.18.E1] || <math qid="Q4570">\WhittakerconfhyperM{0}{\frac{1}{2}}@{2z} = 2\sinh@@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperM{0}{\frac{1}{2}}@{2z} = 2\sinh@@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>WhittakerM(0, (1)/(2), 2*z) = 2*sinh(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerM[0, Divide[1,2], 2*z] == 2*Sinh[z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7] | ||
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| [https://dlmf.nist.gov/13.18.E2 13.18.E2] | | | [https://dlmf.nist.gov/13.18.E2 13.18.E2] || <math qid="Q4571">\WhittakerconfhyperM{\kappa}{\kappa-\frac{1}{2}}@{z} = \WhittakerconfhyperW{\kappa}{\kappa-\frac{1}{2}}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperM{\kappa}{\kappa-\frac{1}{2}}@{z} = \WhittakerconfhyperW{\kappa}{\kappa-\frac{1}{2}}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>WhittakerM(kappa, kappa -(1)/(2), z) = WhittakerW(kappa, kappa -(1)/(2), z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerM[\[Kappa], \[Kappa]-Divide[1,2], z] == WhittakerW[\[Kappa], \[Kappa]-Divide[1,2], z]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 70] | ||
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| [https://dlmf.nist.gov/13.18.E2 13.18.E2] | | | [https://dlmf.nist.gov/13.18.E2 13.18.E2] || <math qid="Q4571">\WhittakerconfhyperW{\kappa}{\kappa-\frac{1}{2}}@{z} = \WhittakerconfhyperW{\kappa}{-\kappa+\frac{1}{2}}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperW{\kappa}{\kappa-\frac{1}{2}}@{z} = \WhittakerconfhyperW{\kappa}{-\kappa+\frac{1}{2}}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>WhittakerW(kappa, kappa -(1)/(2), z) = WhittakerW(kappa, - kappa +(1)/(2), z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerW[\[Kappa], \[Kappa]-Divide[1,2], z] == WhittakerW[\[Kappa], - \[Kappa]+Divide[1,2], z]</syntaxhighlight> || Failure || Successful || Successful [Tested: 70] || Successful [Tested: 70] | ||
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| [https://dlmf.nist.gov/13.18.E2 13.18.E2] | | | [https://dlmf.nist.gov/13.18.E2 13.18.E2] || <math qid="Q4571">\WhittakerconfhyperW{\kappa}{-\kappa+\frac{1}{2}}@{z} = e^{-\frac{1}{2}z}z^{\kappa}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperW{\kappa}{-\kappa+\frac{1}{2}}@{z} = e^{-\frac{1}{2}z}z^{\kappa}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>WhittakerW(kappa, - kappa +(1)/(2), z) = exp(-(1)/(2)*z)*(z)^(kappa)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerW[\[Kappa], - \[Kappa]+Divide[1,2], z] == Exp[-Divide[1,2]*z]*(z)^\[Kappa]</syntaxhighlight> || Failure || Successful || Successful [Tested: 70] || Successful [Tested: 70] | ||
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| [https://dlmf.nist.gov/13.18.E3 13.18.E3] | | | [https://dlmf.nist.gov/13.18.E3 13.18.E3] || <math qid="Q4572">\WhittakerconfhyperM{\kappa}{-\kappa-\frac{1}{2}}@{z} = e^{\frac{1}{2}z}z^{-\kappa}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperM{\kappa}{-\kappa-\frac{1}{2}}@{z} = e^{\frac{1}{2}z}z^{-\kappa}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>WhittakerM(kappa, - kappa -(1)/(2), z) = exp((1)/(2)*z)*(z)^(- kappa)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerM[\[Kappa], - \[Kappa]-Divide[1,2], z] == Exp[Divide[1,2]*z]*(z)^(- \[Kappa])</syntaxhighlight> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [20 / 70]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.012581208495203278, -0.029801099144953658] | ||
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.32783156414330006, -0.2917810845255237] | Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-0.32783156414330006, -0.2917810845255237] | ||
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div> | Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div> | ||
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| [https://dlmf.nist.gov/13.18.E4 13.18.E4] | | | [https://dlmf.nist.gov/13.18.E4 13.18.E4] || <math qid="Q4573">\WhittakerconfhyperM{\mu-\frac{1}{2}}{\mu}@{z} = 2\mu e^{\frac{1}{2}z}z^{\frac{1}{2}-\mu}\incgamma@{2\mu}{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperM{\mu-\frac{1}{2}}{\mu}@{z} = 2\mu e^{\frac{1}{2}z}z^{\frac{1}{2}-\mu}\incgamma@{2\mu}{z}</syntaxhighlight> || <math>\realpart@@{(2\mu)} > 0</math> || <syntaxhighlight lang=mathematica>WhittakerM(mu -(1)/(2), mu, z) = 2*mu*exp((1)/(2)*z)*(z)^((1)/(2)- mu)* GAMMA(2*mu)-GAMMA(2*mu, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerM[\[Mu]-Divide[1,2], \[Mu], z] == 2*\[Mu]*Exp[Divide[1,2]*z]*(z)^(Divide[1,2]- \[Mu])* Gamma[2*\[Mu], 0, z]</syntaxhighlight> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [35 / 35]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.5507089801-1.429327526*I | ||
Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -2.178955063-1.073512810*I | Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -2.178955063-1.073512810*I | ||
Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 35] | Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Successful [Tested: 35] | ||
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| [https://dlmf.nist.gov/13.18.E5 13.18.E5] | | | [https://dlmf.nist.gov/13.18.E5 13.18.E5] || <math qid="Q4574">\WhittakerconfhyperW{\mu-\frac{1}{2}}{\mu}@{z} = e^{\frac{1}{2}z}z^{\frac{1}{2}-\mu}\incGamma@{2\mu}{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperW{\mu-\frac{1}{2}}{\mu}@{z} = e^{\frac{1}{2}z}z^{\frac{1}{2}-\mu}\incGamma@{2\mu}{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>WhittakerW(mu -(1)/(2), mu, z) = exp((1)/(2)*z)*(z)^((1)/(2)- mu)* GAMMA(2*mu, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerW[\[Mu]-Divide[1,2], \[Mu], z] == Exp[Divide[1,2]*z]*(z)^(Divide[1,2]- \[Mu])* Gamma[2*\[Mu], z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 70] | ||
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| [https://dlmf.nist.gov/13.18.E6 13.18.E6] | | | [https://dlmf.nist.gov/13.18.E6 13.18.E6] || <math qid="Q4575">\WhittakerconfhyperM{-\frac{1}{4}}{\frac{1}{4}}@{z^{2}} = \tfrac{1}{2}e^{\frac{1}{2}z^{2}}\sqrt{\pi z}\erf@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperM{-\frac{1}{4}}{\frac{1}{4}}@{z^{2}} = \tfrac{1}{2}e^{\frac{1}{2}z^{2}}\sqrt{\pi z}\erf@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>WhittakerM(-(1)/(4), (1)/(4), (z)^(2)) = (1)/(2)*exp((1)/(2)*(z)^(2))*sqrt(Pi*z)*erf(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerM[-Divide[1,4], Divide[1,4], (z)^(2)] == Divide[1,2]*Exp[Divide[1,2]*(z)^(2)]*Sqrt[Pi*z]*Erf[z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .7978557562-.9869289445*I | ||
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.482664004+.2744150982*I | Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.482664004+.2744150982*I | ||
Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.7978557563768727, -0.986928944338508] | Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.7978557563768727, -0.986928944338508] | ||
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Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div> | Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div> | ||
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| [https://dlmf.nist.gov/13.18.E7 13.18.E7] | | | [https://dlmf.nist.gov/13.18.E7 13.18.E7] || <math qid="Q4576">\WhittakerconfhyperW{-\frac{1}{4}}{+\frac{1}{4}}@{z^{2}} = e^{\frac{1}{2}z^{2}}\sqrt{\pi z}\erfc@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperW{-\frac{1}{4}}{+\frac{1}{4}}@{z^{2}} = e^{\frac{1}{2}z^{2}}\sqrt{\pi z}\erfc@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>WhittakerW(-(1)/(4), +(1)/(4), (z)^(2)) = exp((1)/(2)*(z)^(2))*sqrt(Pi*z)*erfc(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerW[-Divide[1,4], +Divide[1,4], (z)^(2)] == Exp[Divide[1,2]*(z)^(2)]*Sqrt[Pi*z]*Erfc[z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.928317415+.502368653e-1*I | ||
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -2.674168572+2.656547698*I | Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -2.674168572+2.656547698*I | ||
Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.9283174154667808, 0.050236864945780724] | Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.9283174154667808, 0.050236864945780724] | ||
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Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div> | Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div> | ||
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| [https://dlmf.nist.gov/13.18.E7 13.18.E7] | | | [https://dlmf.nist.gov/13.18.E7 13.18.E7] || <math qid="Q4576">\WhittakerconfhyperW{-\frac{1}{4}}{-\frac{1}{4}}@{z^{2}} = e^{\frac{1}{2}z^{2}}\sqrt{\pi z}\erfc@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperW{-\frac{1}{4}}{-\frac{1}{4}}@{z^{2}} = e^{\frac{1}{2}z^{2}}\sqrt{\pi z}\erfc@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>WhittakerW(-(1)/(4), -(1)/(4), (z)^(2)) = exp((1)/(2)*(z)^(2))*sqrt(Pi*z)*erfc(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerW[-Divide[1,4], -Divide[1,4], (z)^(2)] == Exp[Divide[1,2]*(z)^(2)]*Sqrt[Pi*z]*Erfc[z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.928317415+.502368653e-1*I | ||
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -2.674168572+2.656547698*I | Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -2.674168572+2.656547698*I | ||
Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.928317415466781, 0.05023686494578061] | Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.928317415466781, 0.05023686494578061] | ||
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Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div> | Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div> | ||
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| [https://dlmf.nist.gov/13.18.E8 13.18.E8] | | | [https://dlmf.nist.gov/13.18.E8 13.18.E8] || <math qid="Q4577">\WhittakerconfhyperM{0}{\nu}@{2z} = 2^{2\nu+\frac{1}{2}}\EulerGamma@{1+\nu}\sqrt{z}\modBesselI{\nu}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperM{0}{\nu}@{2z} = 2^{2\nu+\frac{1}{2}}\EulerGamma@{1+\nu}\sqrt{z}\modBesselI{\nu}@{z}</syntaxhighlight> || <math>\realpart@@{(1+\nu)} > 0, \realpart@@{(\nu+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>WhittakerM(0, nu, 2*z) = (2)^(2*nu +(1)/(2))* GAMMA(1 + nu)*sqrt(z)*BesselI(nu, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerM[0, \[Nu], 2*z] == (2)^(2*\[Nu]+Divide[1,2])* Gamma[1 + \[Nu]]*Sqrt[z]*BesselI[\[Nu], z]</syntaxhighlight> || Successful || Successful || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 56]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.8586367168171446, -0.6707313588072118] | ||
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -0.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.33759646322286985, -0.8589803343001376] | Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -0.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.33759646322286985, -0.8589803343001376] | ||
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, -0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div> | Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, -0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div> | ||
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| [https://dlmf.nist.gov/13.18.E9 13.18.E9] | | | [https://dlmf.nist.gov/13.18.E9 13.18.E9] || <math qid="Q4578">\WhittakerconfhyperW{0}{\nu}@{2z} = \sqrt{\ifrac{2z}{\pi}}\modBesselK{\nu}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperW{0}{\nu}@{2z} = \sqrt{\ifrac{2z}{\pi}}\modBesselK{\nu}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>WhittakerW(0, nu, 2*z) = sqrt((2*z)/(Pi))*BesselK(nu, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerW[0, \[Nu], 2*z] == Sqrt[Divide[2*z,Pi]]*BesselK[\[Nu], z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 70] | ||
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| [https://dlmf.nist.gov/13.18.E10 13.18.E10] | | | [https://dlmf.nist.gov/13.18.E10 13.18.E10] || <math qid="Q4579">\WhittakerconfhyperW{0}{\frac{1}{3}}@{\tfrac{4}{3}z^{\frac{3}{2}}} = 2\sqrt{\pi}z^{\frac{1}{4}}\AiryAi@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperW{0}{\frac{1}{3}}@{\tfrac{4}{3}z^{\frac{3}{2}}} = 2\sqrt{\pi}z^{\frac{1}{4}}\AiryAi@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>WhittakerW(0, (1)/(3), (4)/(3)*(z)^((3)/(2))) = 2*sqrt(Pi)*(z)^((1)/(4))* AiryAi(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerW[0, Divide[1,3], Divide[4,3]*(z)^(Divide[3,2])] == 2*Sqrt[Pi]*(z)^(Divide[1,4])* AiryAi[z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.246840478+.5335590044*I | ||
Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.24684047859323988, 0.533559004293784] | Test Values: {z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.24684047859323988, 0.533559004293784] | ||
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div> | Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br></div></div> | ||
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| [https://dlmf.nist.gov/13.18.E12 13.18.E12] | | | [https://dlmf.nist.gov/13.18.E12 13.18.E12] || <math qid="Q4581">\WhittakerconfhyperM{-\frac{1}{2}a}{-\frac{1}{4}}@{\tfrac{1}{2}z^{2}} = 2^{\frac{1}{2}a-1}\EulerGamma@{\tfrac{1}{2}a+\tfrac{3}{4}}\sqrt{\ifrac{z}{\pi}}\*\left(\paraU@{a}{z}+\paraU@{a}{-z}\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperM{-\frac{1}{2}a}{-\frac{1}{4}}@{\tfrac{1}{2}z^{2}} = 2^{\frac{1}{2}a-1}\EulerGamma@{\tfrac{1}{2}a+\tfrac{3}{4}}\sqrt{\ifrac{z}{\pi}}\*\left(\paraU@{a}{z}+\paraU@{a}{-z}\right)</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}a+\tfrac{3}{4})} > 0</math> || <syntaxhighlight lang=mathematica>WhittakerM(-(1)/(2)*a, -(1)/(4), (1)/(2)*(z)^(2)) = (2)^((1)/(2)*a - 1)* GAMMA((1)/(2)*a +(3)/(4))*sqrt((z)/(Pi))*(CylinderU(a, z)+ CylinderU(a, - z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerM[-Divide[1,2]*a, -Divide[1,4], Divide[1,2]*(z)^(2)] == (2)^(Divide[1,2]*a - 1)* Gamma[Divide[1,2]*a +Divide[3,4]]*Sqrt[Divide[z,Pi]]*(ParabolicCylinderD[- 1/2 -(a), z]+ ParabolicCylinderD[- 1/2 -(a), - z])</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [8 / 28]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.4546011384-.8349579092*I | ||
Test Values: {a = 3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .58169427e-2+1.789104086*I | Test Values: {a = 3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .58169427e-2+1.789104086*I | ||
Test Values: {a = 3/2, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [8 / 28]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.454601138107828, -0.8349579095614801] | Test Values: {a = 3/2, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [8 / 28]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.454601138107828, -0.8349579095614801] | ||
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Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div> | Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div> | ||
|- | |- | ||
| [https://dlmf.nist.gov/13.18.E13 13.18.E13] | | | [https://dlmf.nist.gov/13.18.E13 13.18.E13] || <math qid="Q4582">\WhittakerconfhyperM{-\frac{1}{2}a}{\frac{1}{4}}@{\tfrac{1}{2}z^{2}} = 2^{\frac{1}{2}a-2}\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{4}}\sqrt{\ifrac{z}{\pi}}\*\left(\paraU@{a}{-z}-\paraU@{a}{z}\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperM{-\frac{1}{2}a}{\frac{1}{4}}@{\tfrac{1}{2}z^{2}} = 2^{\frac{1}{2}a-2}\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{4}}\sqrt{\ifrac{z}{\pi}}\*\left(\paraU@{a}{-z}-\paraU@{a}{z}\right)</syntaxhighlight> || <math>\realpart@@{(\tfrac{1}{2}a+\tfrac{1}{4})} > 0</math> || <syntaxhighlight lang=mathematica>WhittakerM(-(1)/(2)*a, (1)/(4), (1)/(2)*(z)^(2)) = (2)^((1)/(2)*a - 2)* GAMMA((1)/(2)*a +(1)/(4))*sqrt((z)/(Pi))*(CylinderU(a, - z)- CylinderU(a, z))</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerM[-Divide[1,2]*a, Divide[1,4], Divide[1,2]*(z)^(2)] == (2)^(Divide[1,2]*a - 2)* Gamma[Divide[1,2]*a +Divide[1,4]]*Sqrt[Divide[z,Pi]]*(ParabolicCylinderD[- 1/2 -(a), - z]- ParabolicCylinderD[- 1/2 -(a), z])</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .3997621251-.6252084121*I | ||
Test Values: {a = 3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .9306149059+.2046923958*I | Test Values: {a = 3/2, z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .9306149059+.2046923958*I | ||
Test Values: {a = 3/2, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.3997621252402044, -0.6252084117529283] | Test Values: {a = 3/2, z = -1/2*3^(1/2)-1/2*I}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.3997621252402044, -0.6252084117529283] | ||
Line 72: | Line 72: | ||
Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div> | Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div> | ||
|- | |- | ||
| [https://dlmf.nist.gov/13.18.E14 13.18.E14] | | | [https://dlmf.nist.gov/13.18.E14 13.18.E14] || <math qid="Q4583">\WhittakerconfhyperM{\frac{1}{4}+n}{-\frac{1}{4}}@{z^{2}} = (-1)^{n}\frac{n!}{(2n)!}e^{-\frac{1}{2}z^{2}}\sqrt{z}\HermitepolyH{2n}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperM{\frac{1}{4}+n}{-\frac{1}{4}}@{z^{2}} = (-1)^{n}\frac{n!}{(2n)!}e^{-\frac{1}{2}z^{2}}\sqrt{z}\HermitepolyH{2n}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>WhittakerM((1)/(4)+ n, -(1)/(4), (z)^(2)) = (- 1)^(n)*(factorial(n))/(factorial(2*n))*exp(-(1)/(2)*(z)^(2))*sqrt(z)*HermiteH(2*n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerM[Divide[1,4]+ n, -Divide[1,4], (z)^(2)] == (- 1)^(n)*Divide[(n)!,(2*n)!]*Exp[-Divide[1,2]*(z)^(2)]*Sqrt[z]*HermiteH[2*n, z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 4.741276300-.776142297*I | ||
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 9.155588595+2.115036937*I | Test Values: {z = -1/2+1/2*I*3^(1/2), n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 9.155588595+2.115036937*I | ||
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[4.741276296912009, -0.7761422976118018] | Test Values: {z = -1/2+1/2*I*3^(1/2), n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[4.741276296912009, -0.7761422976118018] | ||
Line 78: | Line 78: | ||
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div> | Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div> | ||
|- | |- | ||
| [https://dlmf.nist.gov/13.18.E15 13.18.E15] | | | [https://dlmf.nist.gov/13.18.E15 13.18.E15] || <math qid="Q4584">\WhittakerconfhyperM{\frac{3}{4}+n}{\frac{1}{4}}@{z^{2}} = (-1)^{n}\frac{n!}{(2n+1)!}\frac{e^{-\frac{1}{2}z^{2}}\sqrt{z}}{2}\HermitepolyH{2n+1}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperM{\frac{3}{4}+n}{\frac{1}{4}}@{z^{2}} = (-1)^{n}\frac{n!}{(2n+1)!}\frac{e^{-\frac{1}{2}z^{2}}\sqrt{z}}{2}\HermitepolyH{2n+1}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>WhittakerM((3)/(4)+ n, (1)/(4), (z)^(2)) = (- 1)^(n)*(factorial(n))/(factorial(2*n + 1))*(exp(-(1)/(2)*(z)^(2))*sqrt(z))/(2)*HermiteH(2*n + 1, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerM[Divide[3,4]+ n, Divide[1,4], (z)^(2)] == (- 1)^(n)*Divide[(n)!,(2*n + 1)!]*Divide[Exp[-Divide[1,2]*(z)^(2)]*Sqrt[z],2]*HermiteH[2*n + 1, z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 2.634248102+.148339259*I | ||
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.481689250+1.400565410*I | Test Values: {z = -1/2+1/2*I*3^(1/2), n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.481689250+1.400565410*I | ||
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[2.6342480998741933, 0.14833925882834587] | Test Values: {z = -1/2+1/2*I*3^(1/2), n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[2.6342480998741933, 0.14833925882834587] | ||
Line 84: | Line 84: | ||
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div> | Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div> | ||
|- | |- | ||
| [https://dlmf.nist.gov/13.18.E16 13.18.E16] | | | [https://dlmf.nist.gov/13.18.E16 13.18.E16] || <math qid="Q4585">\WhittakerconfhyperW{\frac{1}{4}+\frac{1}{2}n}{\frac{1}{4}}@{z^{2}} = 2^{-n}e^{-\frac{1}{2}z^{2}}\sqrt{z}\HermitepolyH{n}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperW{\frac{1}{4}+\frac{1}{2}n}{\frac{1}{4}}@{z^{2}} = 2^{-n}e^{-\frac{1}{2}z^{2}}\sqrt{z}\HermitepolyH{n}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>WhittakerW((1)/(4)+(1)/(2)*n, (1)/(4), (z)^(2)) = (2)^(- n)* exp(-(1)/(2)*(z)^(2))*sqrt(z)*HermiteH(n, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerW[Divide[1,4]+Divide[1,2]*n, Divide[1,4], (z)^(2)] == (2)^(- n)* Exp[-Divide[1,2]*(z)^(2)]*Sqrt[z]*HermiteH[n, z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.704303716-.6267307130*I | ||
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -2.370638149+.3880711488*I | Test Values: {z = -1/2+1/2*I*3^(1/2), n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -2.370638149+.3880711488*I | ||
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.7043037156649337, -0.6267307126437623] | Test Values: {z = -1/2+1/2*I*3^(1/2), n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[1.7043037156649337, -0.6267307126437623] | ||
Line 90: | Line 90: | ||
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div> | Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div> | ||
|- | |- | ||
| [https://dlmf.nist.gov/13.18.E17 13.18.E17] | | | [https://dlmf.nist.gov/13.18.E17 13.18.E17] || <math qid="Q4586">\WhittakerconfhyperW{\frac{1}{2}\alpha+\frac{1}{2}+n}{\frac{1}{2}\alpha}@{z} = (-1)^{n}\Pochhammersym{\alpha+1}{n}\WhittakerconfhyperM{\frac{1}{2}\alpha+\frac{1}{2}+n}{\frac{1}{2}\alpha}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperW{\frac{1}{2}\alpha+\frac{1}{2}+n}{\frac{1}{2}\alpha}@{z} = (-1)^{n}\Pochhammersym{\alpha+1}{n}\WhittakerconfhyperM{\frac{1}{2}\alpha+\frac{1}{2}+n}{\frac{1}{2}\alpha}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>WhittakerW((1)/(2)*alpha +(1)/(2)+ n, (1)/(2)*alpha, z) = (- 1)^(n)* pochhammer(alpha + 1, n)*WhittakerM((1)/(2)*alpha +(1)/(2)+ n, (1)/(2)*alpha, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerW[Divide[1,2]*\[Alpha]+Divide[1,2]+ n, Divide[1,2]*\[Alpha], z] == (- 1)^(n)* Pochhammer[\[Alpha]+ 1, n]*WhittakerM[Divide[1,2]*\[Alpha]+Divide[1,2]+ n, Divide[1,2]*\[Alpha], z]</syntaxhighlight> || Failure || Failure || Successful [Tested: 63] || Successful [Tested: 63] | ||
|- | |- | ||
| [https://dlmf.nist.gov/13.18.E17 13.18.E17] | | | [https://dlmf.nist.gov/13.18.E17 13.18.E17] || <math qid="Q4586">(-1)^{n}\Pochhammersym{\alpha+1}{n}\WhittakerconfhyperM{\frac{1}{2}\alpha+\frac{1}{2}+n}{\frac{1}{2}\alpha}@{z} = (-1)^{n}n!e^{-\frac{1}{2}z}z^{\frac{1}{2}\alpha+\frac{1}{2}}\LaguerrepolyL[\alpha]{n}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(-1)^{n}\Pochhammersym{\alpha+1}{n}\WhittakerconfhyperM{\frac{1}{2}\alpha+\frac{1}{2}+n}{\frac{1}{2}\alpha}@{z} = (-1)^{n}n!e^{-\frac{1}{2}z}z^{\frac{1}{2}\alpha+\frac{1}{2}}\LaguerrepolyL[\alpha]{n}@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(- 1)^(n)* pochhammer(alpha + 1, n)*WhittakerM((1)/(2)*alpha +(1)/(2)+ n, (1)/(2)*alpha, z) = (- 1)^(n)* factorial(n)*exp(-(1)/(2)*z)*(z)^((1)/(2)*alpha +(1)/(2))* LaguerreL(n, alpha, z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(- 1)^(n)* Pochhammer[\[Alpha]+ 1, n]*WhittakerM[Divide[1,2]*\[Alpha]+Divide[1,2]+ n, Divide[1,2]*\[Alpha], z] == (- 1)^(n)* (n)!*Exp[-Divide[1,2]*z]*(z)^(Divide[1,2]*\[Alpha]+Divide[1,2])* LaguerreL[n, \[Alpha], z]</syntaxhighlight> || Missing Macro Error || Successful || Skip - symbolical successful subtest || Successful [Tested: 63] | ||
|} | |} | ||
</div> | </div> |
Latest revision as of 11:34, 28 June 2021
DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
---|---|---|---|---|---|---|---|---|
13.18.E1 | \WhittakerconfhyperM{0}{\frac{1}{2}}@{2z} = 2\sinh@@{z} |
|
WhittakerM(0, (1)/(2), 2*z) = 2*sinh(z)
|
WhittakerM[0, Divide[1,2], 2*z] == 2*Sinh[z]
|
Successful | Successful | - | Successful [Tested: 7] |
13.18.E2 | \WhittakerconfhyperM{\kappa}{\kappa-\frac{1}{2}}@{z} = \WhittakerconfhyperW{\kappa}{\kappa-\frac{1}{2}}@{z} |
|
WhittakerM(kappa, kappa -(1)/(2), z) = WhittakerW(kappa, kappa -(1)/(2), z)
|
WhittakerM[\[Kappa], \[Kappa]-Divide[1,2], z] == WhittakerW[\[Kappa], \[Kappa]-Divide[1,2], z]
|
Successful | Successful | Skip - symbolical successful subtest | Successful [Tested: 70] |
13.18.E2 | \WhittakerconfhyperW{\kappa}{\kappa-\frac{1}{2}}@{z} = \WhittakerconfhyperW{\kappa}{-\kappa+\frac{1}{2}}@{z} |
|
WhittakerW(kappa, kappa -(1)/(2), z) = WhittakerW(kappa, - kappa +(1)/(2), z)
|
WhittakerW[\[Kappa], \[Kappa]-Divide[1,2], z] == WhittakerW[\[Kappa], - \[Kappa]+Divide[1,2], z]
|
Failure | Successful | Successful [Tested: 70] | Successful [Tested: 70] |
13.18.E2 | \WhittakerconfhyperW{\kappa}{-\kappa+\frac{1}{2}}@{z} = e^{-\frac{1}{2}z}z^{\kappa} |
|
WhittakerW(kappa, - kappa +(1)/(2), z) = exp(-(1)/(2)*z)*(z)^(kappa)
|
WhittakerW[\[Kappa], - \[Kappa]+Divide[1,2], z] == Exp[-Divide[1,2]*z]*(z)^\[Kappa]
|
Failure | Successful | Successful [Tested: 70] | Successful [Tested: 70] |
13.18.E3 | \WhittakerconfhyperM{\kappa}{-\kappa-\frac{1}{2}}@{z} = e^{\frac{1}{2}z}z^{-\kappa} |
|
WhittakerM(kappa, - kappa -(1)/(2), z) = exp((1)/(2)*z)*(z)^(- kappa)
|
WhittakerM[\[Kappa], - \[Kappa]-Divide[1,2], z] == Exp[Divide[1,2]*z]*(z)^(- \[Kappa])
|
Successful | Successful | - | Failed [20 / 70]
Result: Complex[-0.012581208495203278, -0.029801099144953658]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, 1.5]}
Result: Complex[-0.32783156414330006, -0.2917810845255237]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[κ, 0.5]}
... skip entries to safe data |
13.18.E4 | \WhittakerconfhyperM{\mu-\frac{1}{2}}{\mu}@{z} = 2\mu e^{\frac{1}{2}z}z^{\frac{1}{2}-\mu}\incgamma@{2\mu}{z} |
WhittakerM(mu -(1)/(2), mu, z) = 2*mu*exp((1)/(2)*z)*(z)^((1)/(2)- mu)* GAMMA(2*mu)-GAMMA(2*mu, z)
|
WhittakerM[\[Mu]-Divide[1,2], \[Mu], z] == 2*\[Mu]*Exp[Divide[1,2]*z]*(z)^(Divide[1,2]- \[Mu])* Gamma[2*\[Mu], 0, z]
|
Failure | Successful | Failed [35 / 35] Result: -.5507089801-1.429327526*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}
Result: -2.178955063-1.073512810*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}
... skip entries to safe data |
Successful [Tested: 35] | |
13.18.E5 | \WhittakerconfhyperW{\mu-\frac{1}{2}}{\mu}@{z} = e^{\frac{1}{2}z}z^{\frac{1}{2}-\mu}\incGamma@{2\mu}{z} |
|
WhittakerW(mu -(1)/(2), mu, z) = exp((1)/(2)*z)*(z)^((1)/(2)- mu)* GAMMA(2*mu, z)
|
WhittakerW[\[Mu]-Divide[1,2], \[Mu], z] == Exp[Divide[1,2]*z]*(z)^(Divide[1,2]- \[Mu])* Gamma[2*\[Mu], z]
|
Successful | Successful | - | Successful [Tested: 70] |
13.18.E6 | \WhittakerconfhyperM{-\frac{1}{4}}{\frac{1}{4}}@{z^{2}} = \tfrac{1}{2}e^{\frac{1}{2}z^{2}}\sqrt{\pi z}\erf@{z} |
|
WhittakerM(-(1)/(4), (1)/(4), (z)^(2)) = (1)/(2)*exp((1)/(2)*(z)^(2))*sqrt(Pi*z)*erf(z)
|
WhittakerM[-Divide[1,4], Divide[1,4], (z)^(2)] == Divide[1,2]*Exp[Divide[1,2]*(z)^(2)]*Sqrt[Pi*z]*Erf[z]
|
Failure | Failure | Failed [2 / 7] Result: .7978557562-.9869289445*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}
Result: 1.482664004+.2744150982*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}
|
Failed [2 / 7]
Result: Complex[0.7978557563768727, -0.986928944338508]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
Result: Complex[1.4826640039189691, 0.2744150979001404]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
|
13.18.E7 | \WhittakerconfhyperW{-\frac{1}{4}}{+\frac{1}{4}}@{z^{2}} = e^{\frac{1}{2}z^{2}}\sqrt{\pi z}\erfc@{z} |
|
WhittakerW(-(1)/(4), +(1)/(4), (z)^(2)) = exp((1)/(2)*(z)^(2))*sqrt(Pi*z)*erfc(z)
|
WhittakerW[-Divide[1,4], +Divide[1,4], (z)^(2)] == Exp[Divide[1,2]*(z)^(2)]*Sqrt[Pi*z]*Erfc[z]
|
Failure | Failure | Failed [2 / 7] Result: -1.928317415+.502368653e-1*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}
Result: -2.674168572+2.656547698*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}
|
Failed [2 / 7]
Result: Complex[-1.9283174154667808, 0.050236864945780724]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
Result: Complex[-2.6741685713500765, 2.656547698651725]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
|
13.18.E7 | \WhittakerconfhyperW{-\frac{1}{4}}{-\frac{1}{4}}@{z^{2}} = e^{\frac{1}{2}z^{2}}\sqrt{\pi z}\erfc@{z} |
|
WhittakerW(-(1)/(4), -(1)/(4), (z)^(2)) = exp((1)/(2)*(z)^(2))*sqrt(Pi*z)*erfc(z)
|
WhittakerW[-Divide[1,4], -Divide[1,4], (z)^(2)] == Exp[Divide[1,2]*(z)^(2)]*Sqrt[Pi*z]*Erfc[z]
|
Failure | Failure | Failed [2 / 7] Result: -1.928317415+.502368653e-1*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}
Result: -2.674168572+2.656547698*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}
|
Failed [2 / 7]
Result: Complex[-1.928317415466781, 0.05023686494578061]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
Result: Complex[-2.674168571350077, 2.6565476986517247]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
|
13.18.E8 | \WhittakerconfhyperM{0}{\nu}@{2z} = 2^{2\nu+\frac{1}{2}}\EulerGamma@{1+\nu}\sqrt{z}\modBesselI{\nu}@{z} |
WhittakerM(0, nu, 2*z) = (2)^(2*nu +(1)/(2))* GAMMA(1 + nu)*sqrt(z)*BesselI(nu, z)
|
WhittakerM[0, \[Nu], 2*z] == (2)^(2*\[Nu]+Divide[1,2])* Gamma[1 + \[Nu]]*Sqrt[z]*BesselI[\[Nu], z]
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Successful | Successful | - | Failed [7 / 56]
Result: Complex[-0.8586367168171446, -0.6707313588072118]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, -0.5]}
Result: Complex[0.33759646322286985, -0.8589803343001376]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[ν, -0.5]}
... skip entries to safe data | |
13.18.E9 | \WhittakerconfhyperW{0}{\nu}@{2z} = \sqrt{\ifrac{2z}{\pi}}\modBesselK{\nu}@{z} |
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WhittakerW(0, nu, 2*z) = sqrt((2*z)/(Pi))*BesselK(nu, z)
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WhittakerW[0, \[Nu], 2*z] == Sqrt[Divide[2*z,Pi]]*BesselK[\[Nu], z]
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Successful | Successful | - | Successful [Tested: 70] |
13.18.E10 | \WhittakerconfhyperW{0}{\frac{1}{3}}@{\tfrac{4}{3}z^{\frac{3}{2}}} = 2\sqrt{\pi}z^{\frac{1}{4}}\AiryAi@{z} |
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WhittakerW(0, (1)/(3), (4)/(3)*(z)^((3)/(2))) = 2*sqrt(Pi)*(z)^((1)/(4))* AiryAi(z)
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WhittakerW[0, Divide[1,3], Divide[4,3]*(z)^(Divide[3,2])] == 2*Sqrt[Pi]*(z)^(Divide[1,4])* AiryAi[z]
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Failure | Failure | Failed [1 / 7] Result: -.246840478+.5335590044*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}
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Failed [1 / 7]
Result: Complex[-0.24684047859323988, 0.533559004293784]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
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13.18.E12 | \WhittakerconfhyperM{-\frac{1}{2}a}{-\frac{1}{4}}@{\tfrac{1}{2}z^{2}} = 2^{\frac{1}{2}a-1}\EulerGamma@{\tfrac{1}{2}a+\tfrac{3}{4}}\sqrt{\ifrac{z}{\pi}}\*\left(\paraU@{a}{z}+\paraU@{a}{-z}\right) |
WhittakerM(-(1)/(2)*a, -(1)/(4), (1)/(2)*(z)^(2)) = (2)^((1)/(2)*a - 1)* GAMMA((1)/(2)*a +(3)/(4))*sqrt((z)/(Pi))*(CylinderU(a, z)+ CylinderU(a, - z))
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WhittakerM[-Divide[1,2]*a, -Divide[1,4], Divide[1,2]*(z)^(2)] == (2)^(Divide[1,2]*a - 1)* Gamma[Divide[1,2]*a +Divide[3,4]]*Sqrt[Divide[z,Pi]]*(ParabolicCylinderD[- 1/2 -(a), z]+ ParabolicCylinderD[- 1/2 -(a), - z])
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Failure | Failure | Failed [8 / 28] Result: -.4546011384-.8349579092*I
Test Values: {a = 3/2, z = -1/2+1/2*I*3^(1/2)}
Result: .58169427e-2+1.789104086*I
Test Values: {a = 3/2, z = -1/2*3^(1/2)-1/2*I}
... skip entries to safe data |
Failed [8 / 28]
Result: Complex[-0.454601138107828, -0.8349579095614801]
Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
Result: Complex[0.005816942543956816, 1.7891040854776739]
Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
... skip entries to safe data | |
13.18.E13 | \WhittakerconfhyperM{-\frac{1}{2}a}{\frac{1}{4}}@{\tfrac{1}{2}z^{2}} = 2^{\frac{1}{2}a-2}\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{4}}\sqrt{\ifrac{z}{\pi}}\*\left(\paraU@{a}{-z}-\paraU@{a}{z}\right) |
WhittakerM(-(1)/(2)*a, (1)/(4), (1)/(2)*(z)^(2)) = (2)^((1)/(2)*a - 2)* GAMMA((1)/(2)*a +(1)/(4))*sqrt((z)/(Pi))*(CylinderU(a, - z)- CylinderU(a, z))
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WhittakerM[-Divide[1,2]*a, Divide[1,4], Divide[1,2]*(z)^(2)] == (2)^(Divide[1,2]*a - 2)* Gamma[Divide[1,2]*a +Divide[1,4]]*Sqrt[Divide[z,Pi]]*(ParabolicCylinderD[- 1/2 -(a), - z]- ParabolicCylinderD[- 1/2 -(a), z])
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Failure | Failure | Failed [6 / 21] Result: .3997621251-.6252084121*I
Test Values: {a = 3/2, z = -1/2+1/2*I*3^(1/2)}
Result: .9306149059+.2046923958*I
Test Values: {a = 3/2, z = -1/2*3^(1/2)-1/2*I}
... skip entries to safe data |
Failed [6 / 21]
Result: Complex[0.3997621252402044, -0.6252084117529283]
Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
Result: Complex[0.9306149056064967, 0.20469239560568858]
Test Values: {Rule[a, 1.5], Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
... skip entries to safe data | |
13.18.E14 | \WhittakerconfhyperM{\frac{1}{4}+n}{-\frac{1}{4}}@{z^{2}} = (-1)^{n}\frac{n!}{(2n)!}e^{-\frac{1}{2}z^{2}}\sqrt{z}\HermitepolyH{2n}@{z} |
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WhittakerM((1)/(4)+ n, -(1)/(4), (z)^(2)) = (- 1)^(n)*(factorial(n))/(factorial(2*n))*exp(-(1)/(2)*(z)^(2))*sqrt(z)*HermiteH(2*n, z)
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WhittakerM[Divide[1,4]+ n, -Divide[1,4], (z)^(2)] == (- 1)^(n)*Divide[(n)!,(2*n)!]*Exp[-Divide[1,2]*(z)^(2)]*Sqrt[z]*HermiteH[2*n, z]
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Failure | Failure | Failed [6 / 21] Result: 4.741276300-.776142297*I
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 1}
Result: 9.155588595+2.115036937*I
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 2}
... skip entries to safe data |
Failed [6 / 21]
Result: Complex[4.741276296912009, -0.7761422976118018]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
Result: Complex[9.15558858680754, 2.115036935310196]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
13.18.E15 | \WhittakerconfhyperM{\frac{3}{4}+n}{\frac{1}{4}}@{z^{2}} = (-1)^{n}\frac{n!}{(2n+1)!}\frac{e^{-\frac{1}{2}z^{2}}\sqrt{z}}{2}\HermitepolyH{2n+1}@{z} |
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WhittakerM((3)/(4)+ n, (1)/(4), (z)^(2)) = (- 1)^(n)*(factorial(n))/(factorial(2*n + 1))*(exp(-(1)/(2)*(z)^(2))*sqrt(z))/(2)*HermiteH(2*n + 1, z)
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WhittakerM[Divide[3,4]+ n, Divide[1,4], (z)^(2)] == (- 1)^(n)*Divide[(n)!,(2*n + 1)!]*Divide[Exp[-Divide[1,2]*(z)^(2)]*Sqrt[z],2]*HermiteH[2*n + 1, z]
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Failure | Failure | Failed [6 / 21] Result: 2.634248102+.148339259*I
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 1}
Result: 3.481689250+1.400565410*I
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 2}
... skip entries to safe data |
Failed [6 / 21]
Result: Complex[2.6342480998741933, 0.14833925882834587]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
Result: Complex[3.4816892469231746, 1.4005654089276338]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
13.18.E16 | \WhittakerconfhyperW{\frac{1}{4}+\frac{1}{2}n}{\frac{1}{4}}@{z^{2}} = 2^{-n}e^{-\frac{1}{2}z^{2}}\sqrt{z}\HermitepolyH{n}@{z} |
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WhittakerW((1)/(4)+(1)/(2)*n, (1)/(4), (z)^(2)) = (2)^(- n)* exp(-(1)/(2)*(z)^(2))*sqrt(z)*HermiteH(n, z)
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WhittakerW[Divide[1,4]+Divide[1,2]*n, Divide[1,4], (z)^(2)] == (2)^(- n)* Exp[-Divide[1,2]*(z)^(2)]*Sqrt[z]*HermiteH[n, z]
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Failure | Failure | Failed [6 / 21] Result: 1.704303716-.6267307130*I
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 1}
Result: -2.370638149+.3880711488*I
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 2}
... skip entries to safe data |
Failed [6 / 21]
Result: Complex[1.7043037156649337, -0.6267307126437623]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
Result: Complex[-2.370638148456005, 0.388071148805901]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
13.18.E17 | \WhittakerconfhyperW{\frac{1}{2}\alpha+\frac{1}{2}+n}{\frac{1}{2}\alpha}@{z} = (-1)^{n}\Pochhammersym{\alpha+1}{n}\WhittakerconfhyperM{\frac{1}{2}\alpha+\frac{1}{2}+n}{\frac{1}{2}\alpha}@{z} |
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WhittakerW((1)/(2)*alpha +(1)/(2)+ n, (1)/(2)*alpha, z) = (- 1)^(n)* pochhammer(alpha + 1, n)*WhittakerM((1)/(2)*alpha +(1)/(2)+ n, (1)/(2)*alpha, z)
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WhittakerW[Divide[1,2]*\[Alpha]+Divide[1,2]+ n, Divide[1,2]*\[Alpha], z] == (- 1)^(n)* Pochhammer[\[Alpha]+ 1, n]*WhittakerM[Divide[1,2]*\[Alpha]+Divide[1,2]+ n, Divide[1,2]*\[Alpha], z]
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Failure | Failure | Successful [Tested: 63] | Successful [Tested: 63] |
13.18.E17 | (-1)^{n}\Pochhammersym{\alpha+1}{n}\WhittakerconfhyperM{\frac{1}{2}\alpha+\frac{1}{2}+n}{\frac{1}{2}\alpha}@{z} = (-1)^{n}n!e^{-\frac{1}{2}z}z^{\frac{1}{2}\alpha+\frac{1}{2}}\LaguerrepolyL[\alpha]{n}@{z} |
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(- 1)^(n)* pochhammer(alpha + 1, n)*WhittakerM((1)/(2)*alpha +(1)/(2)+ n, (1)/(2)*alpha, z) = (- 1)^(n)* factorial(n)*exp(-(1)/(2)*z)*(z)^((1)/(2)*alpha +(1)/(2))* LaguerreL(n, alpha, z)
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(- 1)^(n)* Pochhammer[\[Alpha]+ 1, n]*WhittakerM[Divide[1,2]*\[Alpha]+Divide[1,2]+ n, Divide[1,2]*\[Alpha], z] == (- 1)^(n)* (n)!*Exp[-Divide[1,2]*z]*(z)^(Divide[1,2]*\[Alpha]+Divide[1,2])* LaguerreL[n, \[Alpha], z]
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Missing Macro Error | Successful | Skip - symbolical successful subtest | Successful [Tested: 63] |