坎寧安函數

坎寧安函數又稱為皮爾遜-坎寧安函數（Pearson-Cunningham function)是英國數學家坎寧安在1908年首先研究的特殊函數,^[1]，定義如下^[2]：

${\displaystyle \displaystyle \omega _{m,n}(x)={\frac {e^{-x+\pi i(m/2-n)}}{\Gamma (1+n-m/2)}}U(m/2-n,1+m,x).}$

Cunningham function Maple animation

其中U為特里科米函數。

坎寧安在是在用多變數擴展的埃奇沃斯級數，依機率密度函數的矩來近似機率密度函數時用到坎寧安函數，坎寧安函數和一維或多維常係數的擴散方程有關^[1]

坎寧安函數是下列微分方程的解

${\displaystyle xX''+(x+1+m)X'+(n+{\tfrac {1}{2}}m+1)X.}$

與其他函數的關係

• ${\displaystyle \omega _{m,n}(x)={\frac {exp(-x+(1/2*I)*\pi *m-I*\pi *n)*\Gamma (m)*HeunB(-2*m,0,2+4*n,0,{\sqrt {(}}x))}{\Gamma (1+n-(1/2)*m)*x^{m}*\Gamma ((1/2)*m-n)}}}$

${\displaystyle +{\frac {exp(-x+(1/2*I)*Pi*m-I*\pi *n)*\Gamma (-m)*HeunB(2*m,0,2+4*n,0,{\sqrt {(}}x))}{\Gamma (1+n-(1/2)*m)*\Gamma (-(1/2)*m-n)}}}$

• ${\displaystyle \omega _{m,n}={\frac {WhittakerM(0,-1/2,-x+I*\pi *((1/2)*m-n))*exp(-(1/2)*x+(1/2*I)*\pi *((1/2)*m-n))*WhittakerW(1/2+n,(1/2)*m,x)*exp((1/2)*x)}{\Gamma (1+n-(1/2)*m)*x^{(1/2+(1/2)*m)}}}}$

級數展開

${\displaystyle \omega _{0.5,0.5}(x)={(1/80640)*(120960*{\sqrt {(}}2)*\Gamma (3/4)^{2}*{\sqrt {(}}x)-141120*{\sqrt {(}}2)*\Gamma (3/4)^{2}*x^{(}3/2)+77616*{\sqrt {(}}2)*\Gamma (3/4)^{2}*x^{(}5/2)-27720*{\sqrt {(}}2)*\Gamma (3/4)^{2}*x^{(}7/2)+7315*{\sqrt {(}}2)*\Gamma (3/4)^{2}*x^{(}9/2)+(141120*I)*{\sqrt {(}}2)*\Gamma (3/4)^{2}*x^{(}3/2)+(27720*I)*{\sqrt {(}}2)*\Gamma (3/4)^{2}*x^{(}7/2)-(100800*I)*\pi *x-(7315*I)*{\sqrt {(}}2)*\Gamma (3/4)^{2}*x^{(}9/2)-(77616*I)*{\sqrt {(}}2)*\Gamma (3/4)^{2}*x^{(}5/2)-40320*\pi +(75600*I)*\pi *x^{2}+100800*\pi *x+(40320*I)*\pi -75600*\pi *x^{2}-(120960*I)*{\sqrt {(}}2)*\Gamma (3/4)^{2}*{\sqrt {(}}x)+32760*\pi *x^{3}-(32760*I)*\pi *x^{3}-9945*\pi *x^{4}+(9945*I)*\pi *x^{4}+80640*\pi ^{(}3/2)*O(x^{(}9/2))*{\sqrt {(}}x))/(\pi ^{(}3/2)*{\sqrt {(}}x))}}$