功能:第三类椭圆积分

格式:EllipticPi(n,z,k)、EllipticPi(n,k)

说明:

1、第三类不完全椭圆积分EllipticPi(n,z,k)函数等价于Int[(1+n*{Sin<算式解析\sin>}(x)^2)^(-1)*(1-k^2*{Sin<算式解析\sin>}(x)^2)^(-0.5),x,0,z],其中Int[(1+n*{Sin<算式解析\sin>}(x)^2)^(-1)*(1-k^2*{Sin<算式解析\sin>}(x)^2)^(-0.5),x,0,z]表示(1+n*{Sin<算式解析\sin>}(x)^2)^(-1)*(1-k^2*{Sin<算式解析\sin>}(x)^2)^(-0.5)函数对x变量进行0到z定积分.

2、第三类完全椭圆积分EllipticPi(n,k)=EllipticPi(n,π/2,k)

注意:如果c=1/{Sin<算式解析\sin>}(z)^2,那么EllipticPi(n,z,k)={EllipticRF<算式解析\EllipticRF>}（c-1,c-k^2,c)-n/3*{EllipticRJ<算式解析\EllipticRJ>}(c-1,c-k^2,c,c+n)

参考:《NUMERICAL COMPUTATION OF REAL OR COMPLEX ELLIPTIC INTEGRALS》_B.C.CarlsonAmes_Laboratory and Department of Mathematics, Iowa State University,Ames, Iowa 50011-3020, USA

例子:

Ellipticpi(i,i,i)

ans = 0.572019373587981 - 0.849698942824233i