基于Beta函数及其偏导数的广义积分的高精度快速计算

High Precision Fast Calculation of Generalized Integral Based on the Beta Function and Its Partial Derivatives

考虑Beta函数偏导数的计算以及与此相关的广义积分的高精度快速计算问题.首先将Beta函数B(x,y)的定义扩展到整个复平面上,并建立了在整个复平面上Beta函数B(x,y)的偏导数的递推公式.对许多广义积分我们给出Beta函数偏导数的表示形式,因而利用Beta函数的偏导数计算这些广义积分.数值计算表明,算法无论从计算精度还是计算速度,远好于数值积分.另外,得到了B_(p,q)(x,y)存在闭形式的条件,并给出一些广义积分的闭形式.

In this paper, the calculation of the partial derivatives of the Beta function and high precision fast algorithm for generalized integral on the Beta function and its partial derivatives are concerned. The definition of the Beta function B（x, y） is extended to the entire complex plane, and a recursive formula of the partial derivatives Bp,q（x, y） of the Beta function is established in the entire complex plane. Many generalized integrals can be expressed as a form in the partial derivative of the beta function, and thus these generalized integrals can be calculated by use of the partial derivatives of the Beta function. The numerical results show that this algorithm, whether concerning the calculation accuracy or the calculation speed, is superior to the numerical integration. On the other hand, the closed form condition for the partial derivatives Bp,q（x, y） of the Beta function and the closed form of some the generalized integral is obtained.