双重贝塞尔函数积分的数值计算

The numerical integration of dual hankel transformation

双重贝塞尔函数积分由于贝塞尔函数互乘项的强振荡与慢衰减特性而难以应用通常的数值积分算法。本文将被积区间 [0 ,∞ )划分为 [0 ,λ0 ]、[λ0 ,∞ )两部分 ,应用贝塞尔函数的汉克尔函数表述式及后者的大宗量渐近特性 ,区间 [λ0 ,∞ )的双重贝塞尔函数积分可被转化为 Fourier正 (余 )弦变换 ,并可利用各种快速算法对其进行数值计算 ;区间 [0 ,λ0 ]上的双重贝塞尔函数积分的计算可直接应用一般的数值积分算法并能获得较高的计算精度 ;当需大量计算有共同参量的双重贝塞尔函数积分时 ,其计算效率仍显不足。此时 ,可应用贝塞尔函数的导数关系式对 [0 ,λ0 ]内的双重贝塞尔函数积分进行恒等变换 ,再用差商近似导数 ,将其转化为对贝塞尔函数本身的积分 ,而该积分又仅需计算一次。

Due to strong oscillation and slow decay of product term in dual Bessel integration,it is difficult to use ordinary numerical method for the quadrature of dual Bessel.In this paper we divide [0,∞) into [0,λ 0] and [λ 0,∞);In [λ 0,∞),the integration can be written as Fourier cosine and sine transformation by asymptotic expression of Hankel Function and it can be evaluated by fast algorthm.In [0,λ 0],the integration can be computed accurately through direct quadrature.But when we calculate a quantity the related dual Bessel integration with common argument,the algorithm mentioned above seem not to be efficient enough.In this case,according to derivative relation of Bessel function,the constant transformation of the integration in [0,λ 0] can be done and the changed into the integration of the Bessel function itself which can only be calculated only once.This algorithm has improved efficiency of calculation obviously for dual Bessel integration.