Calculations of integrals of products of Bessel functions
Calculations of integrals of products of Bessel functions
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DOI：
10.1090/s0025-5718-67-99149-1
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发表时间：
1967-09
影响因子：
2
通讯作者：
J. E. Kilpatrick;S. Katsura;Y. Inoue
中科院分区：
文献类型：
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作者：
J. E. Kilpatrick;S. Katsura;Y. Inoue
其中 n\, n2, ns 是零或正整数，n\ + n2 + ns 是偶数，a, b 是实数正数。使用符号Ui是为了方便参考。积分 (1.1) 和 (1.2) 以及相关勒让德函数 [7] 的三重和四重乘积的积分，被用于计算统计力学中的维里系数 [3]、[4]、[5]、[6]。常用的数值积分技术，例如辛普森法、高斯法、多项式不定积分法[7]等，当应用于具有振荡被积函数的积分（例如（1.1）和（1.2）中时）效率低下。积分 (1.1) 和 (1.2) 的值可以通过将它们转换为 Mellin-Barnes 积分 [1]、[2] 等或 Meijer 的 G 函数 [2] 来获得，并且应用 [3] 中的一位作者开发的留数微积分可以精确确定积分。由于梅林-巴恩斯被积函数相当复杂，因此必须进行大量扫描才能确定实际的极点。在本文中，扫描过程和这些极残留物的评估是由计算机编程的。
where n\, n2, ns are zero or positive integers, n\ + n2 + ns is even and a, b are real positive numbers. The symbols Ui are used for convenient reference. Integrals (1.1) and (1.2) together with integrals of threefold and fourfold products of associated Legendre functions [7], were used in the calculation of virial coefficients in statistical mechanics [3], [4], [5], [6]. The usual numerical integration techniques such as Simpson's method, Gauss' method, method of indefinite integral of polynomials [7] etc., when applied to integrals with oscillating integrands such as in (1.1) and (1.2) are inefficient. Values of integrals (1.1) and (1.2) can be obtained by transforming them into Mellin-Barnes integrals [1], [2], etc., or Meijer's G-functions [2], and application of the residue calculus as developed by one of the authors in [3] leads to the exact determination of the integrals. As the Mellin-Barnes integrands are rather complicated, one has to do considerable scanning to determine the actual poles. In this paper, the scanning process and the evaluation of the residues at these poles is computer programmed.