We prove a new addition theorem for cylinder functions. The value of an arbitrary zero order cylinder function for an argument x−y with complex-valued x and y is expressed by an infinite series whose summands involve the product of cylinder functions, one of these functions taken at x and the other at y. Other as in the hitherto known addition theorems for cylinder functions, the summands of this series also contain factors that are decreasing powers of x and increasing powers of y, respectively. As a consequence, our addition theorem directly leads to asymptotic expansions at infinity for convolutions of compactly supported functions with zero order cylinder functions.

我们证明了圆柱函数的一个新的加法定理。对于具有复值x和y的参数x - y的任意零阶柱面函数的值由无限级数表示，其和涉及柱面函数的乘积，这些函数之一在x处获取，另一个在y处获取。与迄今为止已知的柱面函数加法定理一样，该级数的和还包含x的幂减小和y的幂增加的因子， 分别。因此，我们的加法定理直接导致紧支持函数与零阶柱面函数的卷积在无穷远处的渐近展开。