查看“︁坎宁安函数”︁的源代码

[[File:Cunningham function Maple animation.gif|thumb|300px|Cunningham function Maple  animation]]
'''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,<ref name=c1908>{{harvtxt|Cunningham|1908}}</ref>，定义如下<ref>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</ref>：

:<math>\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).</math>

其中U为[[合流超几何函数|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 (數學)|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<ref name=c1908/> 

坎宁安函数是下列微分方程的解

:<math> xX''+(x+1+m)X'+(n+\tfrac{1}{2}m+1)X.
</math>
==与其他函数的关系==
*<math>\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)}</math>

<math>  +\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)  }                  </math>

*<math>\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)} }</math>

==级数展开==
<math>\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))}</math>

==腳註==
{{reflist}}

==参考文献==
*{{Citation | last1=Cunningham | first1=E. | title=The ω-Functions, a Class of Normal Functions Occurring in Statistics | jstor=93061 | publisher=The Royal Society | year=1908 | journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character | issn=0950-1207 | volume=81 | issue=548 | pages=310–331 | doi=10.1098/rspa.1908.0085}}

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[[Category:特殊函数]]