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首页> 外文期刊>Computer physics communications >Integrating products of Bessel functions with an additional exponential or rational factor
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Integrating products of Bessel functions with an additional exponential or rational factor

机译：将Bessel函数的乘积与其他指数或有理因数集成

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We provide two MATLAB programs to compute integrals of the form [GRAPHICS] with J(nu i) (x) the Bessel function of the first kind and (real) order vi. The parameter m is a real number such that Sigma(i) nu(i) + m > -1 (to assure integrability near zero), r is real and the numbers c and a(i) are all strictly positive. The program can deliver accurate error estimates. Program summary Program title: BESSELINTR, BESSELINTC Catalogue identifier: AEAH_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEAH_v1-0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 1601 No. of bytes in distributed program, including test data, etc.: 13 161 Distribution format: tar.gz Programming language: Matlab (version >= 6.5), Octave (version >= 2.1.69) Computer. All supporting Matlab or Octave Operating system: All supporting Matlab or Octave RAM: For k Bessel functions our program needs approximately (500 + 140k) double precision variables Classification: 4.11 Nature of problem: The problem consists in integrating an arbitrary product of Bessel functions with an additional rational or exponential factor over a semi-infinite interval. Difficulties arise from the irregular oscillatory behaviour and the possible slow decay of the integrand, which prevents truncation at a finite point. Solution method: The interval of integration is split into a finite and infinite part. The integral over the finite part is computed using Gauss-Legendre quadrature. The integrand on the infinite part is approximated using asymptotic expansions and this approximation is integrated exactly with the aid of the upper incomplete gamma function. In the case where a rational factor is present, this factor is first expanded in a Taylor series around infinity. Restrictions: Some (and eventually all) numerical accuracy is lost when one or more of the parameters r, c, a(i) or vi grow very large, or when r becomes small. Running time: Less than 0.02 s for a simple problem (two Bessel functions, small parameters), A few seconds for a more complex problem (more than six Bessel functions, large parameters), in Matlab 7.4 (R2007a) on a 2.4 GHz AMD Opteron Processor 250.
机译：我们提供了两个MATLAB程序来计算形式为[GRAPHICS]的积分，其中第一类Bessel函数的J（nu i）（x）为（实数）vi。参数m是一个实数，使得Sigma（i）nu（i）+ m> -1（以确保可积性接近零），r为实数，并且数c和a（i）都严格为正。该程序可以提供准确的错误估计。程序摘要程序标题：BESSELINTR，BESSELINTC目录标识符：AEAH_v1_0程序摘要URL：http://cpc.cs.qub.ac.uk/summaries/AEAH_v1-0.html程序可从以下网站获得：CPC程序库，皇后大学，贝尔法斯特， N.爱尔兰许可条款：标准CPC许可，http：//cpc.cs.qub.ac.uk/licence/licence.html分布式程序中的行数，包括测试数据等：1601中的字节数分布式程序，包括测试数据等：13 161分发格式：tar.gz编程语言：Matlab（版本> = 6.5），八度（版本> = 2.1.69）计算机。所有支持Matlab或Octave的操作系统：所有支持Matlab或Octave的RAM：对于k个Bessel函数，我们的程序需要大约（500 + 140k）双精度变量分类：4.11问题的性质：问题在于将Bessel函数的任意乘积与半无限间隔内的附加有理或指数因子。困难来自于不规则的振荡行为以及被积物可能缓慢的衰减，这阻止了在有限点处的截断。解决方法：积分间隔分为有限和无限部分。使用Gauss-Legendre正交计算有限部分上的积分。使用渐近展开来逼近无穷大部分上的被积，并借助上级不完全伽马函数精确地将这一近似积分。在存在有理因数的情况下，该因数首先在无穷大附近的泰勒级数中展开。限制：当参数r，c，a（i）或vi中的一个或多个变得非常大时，或者当r变小时，会失去一些（最终所有）数值精度。运行时间：在2.4 GHz AMD的Matlab 7.4（R2007a）中，一个简单的问题（两个Bessel函数，小的参数）少于0.02 s，一个更复杂的问题（超过六个Bessel函数，大参数）少于几秒。皓龙处理器250。

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