两个球Bessel函数乘积的符号积分及其附加指数和多项式因子
Symbolic integration of a product of two spherical bessel functions with an additional exponential and polynomial factor
摘要：
我们提出了一个符号计算的软件包，执行积分的形式\int^{\infty}_0 e^{-x/u} x^n j_v（x）j_μ（x）dx其中$j_ν (x)$和$j_μ (x)$表示球面贝塞尔函数的整数阶，与$ν\ge 0$和$μ\ge 0$。对于实参数$u>0$和整数$n$，积分的收敛要求$n+ν+μ\ge 0$。该软件包以最简化的形式提供积分的分析结果。所采用的新的符号方法，使大量的上述形式的积分的计算在传统的数值和Mathematica的蛮力方法所需的时间的一小部分。我们测试的准确性，这样的解析表达式进行比较的结果与数值对应。
英文摘要：
We present a mathematica package that performs the symbolic calculation of integrals of the form \int^{\infty}_0 e^{-x/u} x^n j_ν (x) j_μ (x) dx where $j_ν (x)$ and $j_μ (x)$ denote spherical Bessel functions of integer orders, with $ν\ge 0$ and $μ\ge 0$. With the real parameter $u>0$ and the integer $n$, convergence of the integral requires that $n+ν+μ\ge 0$. The package provides analytical result for the integral in its most simplified form. The novel symbolic method employed enables the calculation of a large number of integrals of the above form in a fraction of the time required for conventional numerical and Mathematica based brute-force methods. We test the accuracy of such analytical expressions by comparing the results with their numerical counterparts.
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