本文提出了一种柱坐标系下亥姆霍兹格林函数的傅里叶分解新方法，它等价于求解一般环源的亥姆霍兹方程。将亥姆霍兹格林函数的傅里叶系数分为半超前+半滞后分量和半超前-半滞后分量。封闭形式的解决方案，给出了这些组件的霍恩函数和Kampe de Feriet函数，分别。与这些二维超几何函数相关联的偏微分方程系统被用来构造两个分量都满足的四阶常微分方程。本文从广义傅里叶系数的积分表示出发，导出了广义傅里叶系数的第二个四阶常微分方程，并证明了这两个微分方程是等价的。各种傅立叶系数的级数解也给出了，主要是在勒让德函数和贝塞尔/汉克尔函数。这些都是来自封闭形式的超几何解决方案或积分表示，或两者兼而有之。数值计算比较不同的方法计算傅立叶系数。
英文摘要：
A new method is presented for Fourier decomposition of the Helmholtz Green Function in cylindrical coordinates, which is equivalent to obtaining the solution of the Helmholtz equation for a general ring source. The Fourier coefficients of the Helmholtz Green function are split into their half advanced+half retarded and half advanced-half retarded components. Closed form solutions are given for these components in terms of a Horn function and a Kampe de Feriet function, respectively. The systems of partial differential equations associated with these two-dimensional hypergeometric functions are used to construct a fourth-order ordinary differential equation which both components satisfy. A second fourth-order ordinary differential equation for the general Fourier coefficent is derived from an integral representation of the coefficient, and both differential equations are shown to be equivalent. Series solutions for the various Fourier coefficients are also given, mostly in terms of Legendre functions and Bessel/Hankel functions. These are derived from the closed form hypergeometric solutions or an integral representation, or both. Numerical calculations comparing different methods of calculating the Fourier coefficients are presented.
