13.24

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
13.24.E1	${\displaystyle{\displaystyle M_{\kappa,\mu}\left(z\right)=\Gamma\left(\kappa+\mu\right)2^{2\kappa+2\mu}z^{\frac{1}{2}-\kappa}\*\sum_{s=0}^{\infty}(-1)^{s}\frac{{\left(2\kappa+2\mu\right)_{s}}{\left(2\kappa\right)_{s}}}{{\left(1+2\mu\right)_{s}}s!}\*\left(\kappa+\mu+s\right)I_{\kappa+\mu+s}\left(\tfrac{1}{2}z\right)}}$ \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}	${\displaystyle{\displaystyle\Re(\kappa+\mu)>0,\Re((\kappa+\mu+s)+k+1)>0}}$	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)	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]	Failure	Failure	Manual Skip!	Skipped - Because timed out
13.24.E2	${\displaystyle{\displaystyle\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}\left(z\right)=2^{2\mu}z^{\mu+\frac{1}{2}}\sum_{s=0}^{\infty}p_{s}^{(\mu)}(z)\left(2\sqrt{\kappa z}\right)^{-2\mu-s}J_{2\mu+s}\left(2\sqrt{\kappa z}\right)}}$ \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}}	${\displaystyle{\displaystyle\Re((2\mu+s)+k+1)>0,\Re(1+2\mu)>0}}$	(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)	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]	Failure	Failure	Skipped - Because timed out	Skipped - Because timed out
13.24.E3	${\displaystyle{\displaystyle\exp\left(-\tfrac{1}{2}z\left(\coth t-\frac{1}{t}\right)\right)\left(\frac{t}{\sinh t}\right)^{1-2\mu}=\sum_{s=0}^{\infty}p_{s}^{(\mu)}(z)\left(-\frac{t}{z}\right)^{s}}}$ \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}		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)	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]	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