在Cap-cyclide坐标中Wangerin函数N_m^n(ν)的高精度数值计算

The high precision numerical calculation of Wangerin function N_m^n(v) based on Cap-cyclide coordinates

在Cap-cyclide坐标中,Wangerin函数Nnm(v)为特征值函数且解析式中包含第一类完全椭圆积分和Jacobi椭圆函数。为实现Wangerin函数Nnm(v)的高精度数值计算,首先利用迭代法对第一类完全椭圆积分进行数值计算,得到的数值解与理论值基本一致;其次利用Jacobi椭圆函数的恒等式实现其数值计算,数值解的有效数字达到了14位以上。基于此,分两个步骤实现Wangerin函数Nnm(v)的高精度数值计算。本文的结论为进一步探讨Wangerin函数的收敛性和稳定性问题提供基础,具有一定的工程实际价值。

In Cap-cyclide coordinates, the Wangerin function Nm^n（v）, which is eigenvalue function, includes complete elliptic integral of the first kind and Jacobi elliptic functions. To get the high precision numerical solutions of Wangerin function Nm^n（v）, firstly, the iterative method is applied to take numerical calculation in complete elliptic integral of the first kind. The numerical results are as accurate as theoretic values. Secondly, we carry on numerical calculation with the identical equations of Jacohi elliptic functions, and the number of numerical solutions~ significant digits is up to 14. Then, based on these results, we procure the high precision numerical solutions of Wangerin function Nm^n（v） upon these two steps. The result provides the theoretical basis for further research on convergence and stability of Wan- gerin function Nm^n（v）, and shows that it is very useful to practical engineering.