View source for 13.24

{{DISPLAYTITLE:Confluent Hypergeometric Functions - 13.24 Series}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
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! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/13.24.E1 13.24.E1] || [[Item:Q4649|<math>\WhittakerconfhyperM{\kappa}{\mu}@{z} = \EulerGamma@{\kappa+\mu}2^{2\kappa+2\mu}z^{\frac{1}{2}-\kappa}\*\sum_{s=0}^{\infty}(-1)^{s}\frac{\Pochhammersym{2\kappa+2\mu}{s}\Pochhammersym{2\kappa}{s}}{\Pochhammersym{1+2\mu}{s}s!}\*\left(\kappa+\mu+s\right)\modBesselI{\kappa+\mu+s}@{\tfrac{1}{2}z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\WhittakerconfhyperM{\kappa}{\mu}@{z} = \EulerGamma@{\kappa+\mu}2^{2\kappa+2\mu}z^{\frac{1}{2}-\kappa}\*\sum_{s=0}^{\infty}(-1)^{s}\frac{\Pochhammersym{2\kappa+2\mu}{s}\Pochhammersym{2\kappa}{s}}{\Pochhammersym{1+2\mu}{s}s!}\*\left(\kappa+\mu+s\right)\modBesselI{\kappa+\mu+s}@{\tfrac{1}{2}z}</syntaxhighlight> || <math>\realpart@@{(\kappa+\mu)} > 0, \realpart@@{((\kappa+\mu+s)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>WhittakerM(kappa, mu, z) = GAMMA(kappa + mu)*(2)^(2*kappa + 2*mu)* (z)^((1)/(2)- kappa)* sum((- 1)^(s)*(pochhammer(2*kappa + 2*mu, s)*pochhammer(2*kappa, s))/(pochhammer(1 + 2*mu, s)*factorial(s))*(kappa + mu + s)*BesselI(kappa + mu + s, (1)/(2)*z), s = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>WhittakerM[\[Kappa], \[Mu], z] == Gamma[\[Kappa]+ \[Mu]]*(2)^(2*\[Kappa]+ 2*\[Mu])* (z)^(Divide[1,2]- \[Kappa])* Sum[(- 1)^(s)*Divide[Pochhammer[2*\[Kappa]+ 2*\[Mu], s]*Pochhammer[2*\[Kappa], s],Pochhammer[1 + 2*\[Mu], s]*(s)!]*(\[Kappa]+ \[Mu]+ s)*BesselI[\[Kappa]+ \[Mu]+ s, Divide[1,2]*z], {s, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Manual Skip! || Skipped - Because timed out
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| [https://dlmf.nist.gov/13.24.E2 13.24.E2] || [[Item:Q4650|<math>\frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{z} = 2^{2\mu}z^{\mu+\frac{1}{2}}\sum_{s=0}^{\infty}p_{s}^{(\mu)}(z)\left(2\sqrt{\kappa z}\right)^{-2\mu-s}\BesselJ{2\mu+s}@{2\sqrt{\kappa z}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{1}{\EulerGamma@{1+2\mu}}\WhittakerconfhyperM{\kappa}{\mu}@{z} = 2^{2\mu}z^{\mu+\frac{1}{2}}\sum_{s=0}^{\infty}p_{s}^{(\mu)}(z)\left(2\sqrt{\kappa z}\right)^{-2\mu-s}\BesselJ{2\mu+s}@{2\sqrt{\kappa z}}</syntaxhighlight> || <math>\realpart@@{((2\mu+s)+k+1)} > 0, \realpart@@{(1+2\mu)} > 0</math> || <syntaxhighlight lang=mathematica>(1)/(GAMMA(1 + 2*mu))*WhittakerM(kappa, mu, z) = (2)^(2*mu)* (z)^(mu +(1)/(2))* sum((p[s])^(mu)(z)*(2*sqrt(kappa*z))^(- 2*mu - s)* BesselJ(2*mu + s, 2*sqrt(kappa*z)), s = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,Gamma[1 + 2*\[Mu]]]*WhittakerM[\[Kappa], \[Mu], z] == (2)^(2*\[Mu])* (z)^(\[Mu]+Divide[1,2])* Sum[(Subscript[p, s])^(\[Mu])[z]*(2*Sqrt[\[Kappa]*z])^(- 2*\[Mu]- s)* BesselJ[2*\[Mu]+ s, 2*Sqrt[\[Kappa]*z]], {s, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/13.24.E3 13.24.E3] || [[Item:Q4651|<math>\exp@{-\tfrac{1}{2}z\left(\coth@@{t}-\frac{1}{t}\right)}\left(\frac{t}{\sinh@@{t}}\right)^{1-2\mu} = \sum_{s=0}^{\infty}p_{s}^{(\mu)}(z)\left(-\frac{t}{z}\right)^{s}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\exp@{-\tfrac{1}{2}z\left(\coth@@{t}-\frac{1}{t}\right)}\left(\frac{t}{\sinh@@{t}}\right)^{1-2\mu} = \sum_{s=0}^{\infty}p_{s}^{(\mu)}(z)\left(-\frac{t}{z}\right)^{s}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>exp(-(1)/(2)*z*(coth(t)-(1)/(t)))*((t)/(sinh(t)))^(1 - 2*mu) = sum((p[s])^(mu)(z)*(-(t)/(z))^(s), s = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[-Divide[1,2]*z*(Coth[t]-Divide[1,t])]*(Divide[t,Sinh[t]])^(1 - 2*\[Mu]) == Sum[(Subscript[p, s])^(\[Mu])[z]*(-Divide[t,z])^(s), {s, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[1.4000146541353637, 0.6933735030866136], Times[-1.0, NSum[Times[Power[Complex[1.299038105676658, -0.7499999999999999], s], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Plus[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], p]
Test Values: {s, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, s], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[1.4000146541353637, 0.6933735030866136], Times[-1.0, NSum[Times[Power[Complex[1.299038105676658, -0.7499999999999999], s], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Power[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], p]
Test Values: {s, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, s], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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