坎宁安函数

坎宁安函数又称为皮尔逊-坎宁安函数（Pearson-Cunningham function)是英国数学家坎宁安在1908年首先研究的特殊函数,^[1]，定义如下^[2]：

Cunningham function Maple animation

\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).

其中U为特里科米函数。

坎宁安在是在用多变数扩展的埃奇沃斯级数，依几率密度函数的矩来近似几率密度函数时用到坎宁安函数，坎宁安函数和一维或多维常系数的扩散方程有关^[1]

坎宁安函数是下列微分方程的解

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

与其他函数的关系

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

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

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

级数展开

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

脚注

^ ^1.0 ^1.1 Cunningham (1908)
^ Cunningham, E. (1908), "The ω-Functions, a Class of Normal Functions Occurring in Statistics", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character (The Royal Society) 81 (548): 310–331, doi:10.1098/rspa.1908.0085, ISSN 0950-1207, JSTOR 93061

参考文献

Cunningham, E., The ω-Functions, a Class of Normal Functions Occurring in Statistics, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character (The Royal Society), 1908, 81 (548): 310–331, ISSN 0950-1207, JSTOR 93061, doi:10.1098/rspa.1908.0085

[c1908-1] 1.0 ^1.1 Cunningham (1908)

[2] Cunningham, E. (1908), "The ω-Functions, a Class of Normal Functions Occurring in Statistics", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character (The Royal Society) 81 (548): 310–331, doi:10.1098/rspa.1908.0085, ISSN 0950-1207, JSTOR 93061

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