Confluent Hypergeometric Functions - 13.24 Series
| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
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| 13.24.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}
\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} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\kappa+\mu)} > 0, \realpart@@{((\kappa+\mu+s)+k+1)} > 0} | 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)
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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]
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Failure | Failure | Manual Skip! | Skipped - Because timed out |
| 13.24.E2 | 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} = 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}}}
\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}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{((2\mu+s)+k+1)} > 0, \realpart@@{(1+2\mu)} > 0} | (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)
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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]
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Failure | Failure | Skipped - Because timed out | Skipped - Because timed out |
| 13.24.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}
\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} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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)
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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]
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Failure | Failure | Skipped - Because timed out | Failed [300 / 300]
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]]]}
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]]]}
... skip entries to safe data |