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,，定义如下： ω m , n ( x ) = e − x + π i ( m / 2 − n ) Γ ( 1 + n − m / 2 ) U ( m / 2 − n , 1 + m , x ) . {\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).} 其中U为特里科米函数。 坎宁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宁安函数，坎宁安函数和一維或多維常係數的擴散方程有關 坎宁安函数是下列微分方程的解 x X ″ + ( x + 1 + m ) X ′ + ( n + 1 2 m + 1 ) X . {\displaystyle xX''+(x+1+m)X'+(n+{\tfrac {1}{2}}m+1)X.} ω m , n ( x ) = e x p ( − x + ( 1 / 2 ∗ I ) ∗ π ∗ m − I ∗ π ∗ n ) ∗ Γ ( m ) ∗ H e u n B ( − 2 ∗ m , 0 , 2 + 4 ∗ n , 0 , ( x ) ) Γ ( 1 + n − ( 1 / 2 ) ∗ m ) ∗ x m ∗ Γ ( ( 1 / 2 ) ∗ m − n ) {\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)}}} + e x p ( − x + ( 1 / 2 ∗ I ) ∗ P i ∗ m − I ∗ π ∗ n ) ∗ Γ ( − m ) ∗ H e u n B ( 2 ∗ m , 0 , 2 + 4 ∗ n , 0 , ( x ) ) Γ ( 1 + n − ( 1 / 2 ) ∗ m ) ∗ Γ ( − ( 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)}}} ω m , n = W h i t t a k e r M ( 0 , − 1 / 2 , − x + I ∗ π ∗ ( ( 1 / 2 ) ∗ m − n ) ) ∗ e x p ( − ( 1 / 2 ) ∗ x + ( 1 / 2 ∗ I ) ∗ π ∗ ( ( 1 / 2 ) ∗ m − n ) ) ∗ W h i t t a k e r W ( 1 / 2 + n , ( 1 / 2 ) ∗ m , x ) ∗ e x p ( ( 1 / 2 ) ∗ x ) Γ ( 1 + n − ( 1 / 2 ) ∗ m ) ∗ x ( 1 / 2 + ( 1 / 2 ) ∗ m ) {\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)}}}} ω 0.5 , 0.5 ( x ) = ( 1 / 80640 ) ∗ ( 120960 ∗ ( 2 ) ∗ Γ ( 3 / 4 ) 2 ∗ ( x ) − 141120 ∗ ( 2 ) ∗ Γ ( 3 / 4 ) 2 ∗ x ( 3 / 2 ) + 77616 ∗ ( 2 ) ∗ Γ ( 3 / 4 ) 2 ∗ x ( 5 / 2 ) − 27720 ∗ ( 2 ) ∗ Γ ( 3 / 4 ) 2 ∗ x ( 7 / 2 ) + 7315 ∗ ( 2 ) ∗ Γ ( 3 / 4 ) 2 ∗ x ( 9 / 2 ) + ( 141120 ∗ I ) ∗ ( 2 ) ∗ Γ ( 3 / 4 ) 2 ∗ x ( 3 / 2 ) + ( 27720 ∗ I ) ∗ ( 2 ) ∗ Γ ( 3 / 4 ) 2 ∗ x ( 7 / 2 ) − ( 100800 ∗ I ) ∗ π ∗ x − ( 7315 ∗ I ) ∗ ( 2 ) ∗ Γ ( 3 / 4 ) 2 ∗ x ( 9 / 2 ) − ( 77616 ∗ I ) ∗ ( 2 ) ∗ Γ ( 3 / 4 ) 2 ∗ x ( 5 / 2 ) − 40320 ∗ π + ( 75600 ∗ I ) ∗ π ∗ x 2 + 100800 ∗ π ∗ x + ( 40320 ∗ I ) ∗ π − 75600 ∗ π ∗ x 2 − ( 120960 ∗ I ) ∗ ( 2 ) ∗ Γ ( 3 / 4 ) 2 ∗ ( x ) + 32760 ∗ π ∗ x 3 − ( 32760 ∗ I ) ∗ π ∗ x 3 − 9945 ∗ π ∗ x 4 + ( 9945 ∗ I ) ∗ π ∗ x 4 + 80640 ∗ π ( 3 / 2 ) ∗ O ( x ( 9 / 2 ) ) ∗ ( x ) ) / ( π ( 3 / 2 ) ∗ ( x ) ) {\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))}} 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 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