# 惠泰克函数

WhittakerM function
Whittaker W function

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

MU

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

## 级数展开

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

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

## 参考文献

1. ^ 王竹溪 郭敦仁 《特殊函数概论》 第291-304页，2000年 北京大学出版社。
2. ^ Frank J. Oliver,NIST Handbook of Mathematical Functions, p395,Cambridge University Press, 2010