惠泰克函数惠泰克函数,惠泰克1904推导合流超几何函数,是下列惠泰克方程的解[1] WhittakerM function Whittaker W function d 2 w d z 2 + ( − 1 4 + κ z + 1 / 4 − μ 2 z 2 ) w = 0. {\displaystyle {\frac {d^{2}w}{dz^{2}}}+\left(-{\frac {1}{4}}+{\frac {\kappa }{z}}+{\frac {1/4-\mu ^{2}}{z^{2}}}\right)w=0.} 此方程在 0 有用正则奇点,在 ∞ 有非正则奇点. 惠泰克方程有两个解[2]M 与 U : M κ , μ ( z ) = exp ( − z / 2 ) z μ + 1 2 M ( μ − κ + 1 2 , 1 + 2 μ ; z ) {\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)} W κ , μ ( z ) = exp ( − z / 2 ) z μ + 1 2 U ( μ − κ + 1 2 , 1 + 2 μ ; z ) . {\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 惠塔克M函数 W h i t t a k e r M = ∑ k = 0 ∞ ( 1 / 2 − a + b ) k ∗ z b + 1 / 2 + k e z / 2 ∗ k ! ∗ ( 1 + 2 b ) k {\displaystyle WhittakerM=\sum _{k=0}^{\infty }{\frac {(1/2-a+b)_{k}*z^{b+1/2+k}}{e^{z/2}*k!*(1+2b)_{k}}}} [ W h i t t a k e r W ( a , b , z ) = ∑ | k 1 = 0 ∞ ( − P i ∗ ( z ( b + 1 / 2 + k 1 ) ∗ Γ ( 1 / 2 − a + b + k 1 ) ∗ Γ ( 1 − 2 ∗ b + k 1 ) − Γ ( 1 / 2 − a − b + k 1 ) ∗ z ( − b + 1 / 2 + k 1 ) ∗ Γ ( k 1 + 1 + 2 ∗ b ) ) / ( G A M M A ( k 1 + 1 ) ∗ G A M M A ( 1 / 2 − a + b ) ∗ G A M M A ( 1 / 2 − a − b ) ∗ s i n ( 2 ∗ P i ∗ b ) ∗ G A M M A ( k 1 + 1 + 2 ∗ b ) ∗ e x p ( ( 1 / 2 ) ∗ z ) ∗ G A M M A ( 1 − 2 ∗ b + k 1 ) ) , k 1 = 0.. i n f i n i t y ) , A n d ( b :: ( N o t ( n o n p o s i n t ) ) , ( 1 / 2 − a + b ) :: ( N o t ( n o n p o s i n t ) ) , ( 1 / 2 − a − b ) :: ( N o t ( n o n p o s i n t ) ) , a b s ( z ) < 1 ) ] {\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 ^ 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。 ^ Frank J. Oliver,NIST Handbook of Mathematical Functions, p395,Cambridge University Press, 2010