惠泰克函數

惠泰克函數，惠泰克1904推導合流超幾何函數，是下列惠泰克方程的解^[1]

${\displaystyle {\frac {d^{2}w}{dz^{2}}}+\left(-{\frac {1}{4}}+{\frac {\kappa }{z}}+{\frac {1/4-\mu ^{2}}{z^{2}}}\right)w=0.}$

WhittakerM function

Whittaker W function

此方程在 0 有用正則奇點，在 ∞ 有非正則奇點.

惠泰克方程有兩個解^[2]

M 與 U ：

${\displaystyle M_{\kappa ,\mu }\left(z\right)=\exp \left(-z/2\right)z^{\mu +{\tfrac {1}{2}}}M\left(\mu -\kappa +{\frac {1}{2}},1+2\mu ;z\right)}$

${\displaystyle W_{\kappa ,\mu }\left(z\right)=\exp \left(-z/2\right)z^{\mu +{\tfrac {1}{2}}}U\left(\mu -\kappa +{\frac {1}{2}},1+2\mu ;z\right).}$

級數展開

惠塔克M函數

${\displaystyle WhittakerM=\sum _{k=0}^{\infty }{\frac {(1/2-a+b)_{k}*z^{b+1/2+k}}{e^{z/2}*k!*(1+2b)_{k}}}}$

${\displaystyle [WhittakerW(a,b,z)=\sum |_{k1=0}^{\infty }(-Pi*(z^{(}b+1/2+_{k}1)*\Gamma (1/2-a+b+_{k}1)*\Gamma (1-2*b+_{k}1)-\Gamma (1/2-a-b+_{k}1)*z^{(}-b+1/2+_{k}1)*\Gamma (_{k}1+1+2*b))/(GAMMA(_{k}1+1)*GAMMA(1/2-a+b)*GAMMA(1/2-a-b)*sin(2*Pi*b)*GAMMA(_{k}1+1+2*b)*exp((1/2)*z)*GAMMA(1-2*b+_{k}1)),_{k}1=0..infinity),And(b::(Not(nonposint)),(1/2-a+b)::(Not(nonposint)),(1/2-a-b)::(Not(nonposint)),abs(z)<1)]}$