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] [2]
Cunningham function Maple animation
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{\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 ，依概率密度函數 的矩 來近似概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系數的擴散方程 有關[1]
坎寧安函數是下列微分方程的解
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{\displaystyle xX''+(x+1+m)X'+(n+{\tfrac {1}{2}}m+1)X.}
與其他函數的關係
編輯
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{\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)}}}
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{\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)}}}
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{\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
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{\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))}}
^ 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
參考文獻
編輯
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 
