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

\frac{d^2w}{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

M 与 U ：

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_{\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).

惠塔克M函数

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

$[WhittakerW(a, b, z) = \sum|_{k1=0}^{\infty}(-Pi*(z^(b+1/2+_k1)*\Gamma(1/2-a+b+_k1)*\Gamma(1-2*b+_k1)-\Gamma(1/2-a-b+_k1)*z^(-b+1/2+_k1)*\Gamma(_k1+1+2*b))/(GAMMA(_k1+1)*GAMMA(1/2-a+b)*GAMMA(1/2-a-b)*sin(2*Pi*b)*GAMMA(_k1+1+2*b)*exp((1/2)*z)*GAMMA(1-2*b+_k1)), _k1 = 0 .. infinity), And(b::(Not(nonposint)), (1/2-a+b)::(Not(nonposint)), (1/2-a-b)::(Not(nonposint)), abs(z) < 1)]$

单词	Whittaker function
释义	Whittaker function 中文百科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 $\frac{d^2w}{dz^2}+\left(-\frac{1}{4}+\frac{\kappa}{z}+\frac{1/4-\mu^2}{z^2}\right)w=0.$ 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 M 与 U ： $M_{\kappa,\mu}\left(z\right) = \exp\left(-z/2\right)z^{\mu+\tfrac{1}{2}}M\left(\mu-\kappa+\frac{1}{2}, 1+2\mu; z\right)$ $W_{\kappa,\mu}\left(z\right) = \exp\left(-z/2\right)z^{\mu+\tfrac{1}{2}}U\left(\mu-\kappa+\frac{1}{2}, 1+2\mu; z\right).$ 惠塔克M函数 $WhittakerM=\sum_{k=0}^{\infty}\frac{(1/2-a+b)_{k}z^{b+1/2+k}}{e^{z/2}k!(1+2b)_k}$ $[WhittakerW(a, b, z) = \sum\|_{k1=0}^{\infty}(-Pi(z^(b+1/2+_k1)\Gamma(1/2-a+b+_k1)\Gamma(1-2b+_k1)-\Gamma(1/2-a-b+_k1)z^(-b+1/2+_k1)\Gamma(_k1+1+2b))/(GAMMA(_k1+1)GAMMA(1/2-a+b)GAMMA(1/2-a-b)sin(2Pib)GAMMA(_k1+1+2b)exp((1/2)z)GAMMA(1-2*b+_k1)), _k1 = 0 .. infinity), And(b::(Not(nonposint)), (1/2-a+b)::(Not(nonposint)), (1/2-a-b)::(Not(nonposint)), abs(z) < 1)]$ 英语百科 Whittaker function 惠泰克函数 In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by Whittaker (1904) to make the formulas involving the solutions more symmetric. More generally, Jacquet (1966, 1967) introduced Whittaker functions of reductive groups over local fields, where the functions studied by Whittaker are essentially the case where the local field is the real numbers and the group is SL₂(R).
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Whittaker function

惠泰克函数

Whittaker function 惠泰克函数