Uniform asymptotic expansions of the modified Bessel function of the third kind of large imaginary order
Uniform asymptotic expansions of the modified Bessel function of the third kind of large imaginary order
复制标题
第三类大虚阶修正贝塞尔函数的一致渐近展开
DOI：
10.1090/s0002-9904-1966-11408-8
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发表时间：
1966
影响因子：
1.3
通讯作者：
C. Balogh
中科院分区：
文献类型：
--
作者：
C. Balogh
修正的贝塞尔方程 dw 1 dw ( v\ (1) + ( l U « 0 dz z dz \ z) 及其特定解 w**Ki»(z)t 具有纯虚数阶的第三类修正贝塞尔函数在脉冲衍射理论中具有根本意义。此外，该函数是 Lebedev 变换的核心 [3]。除了 Friedlander 的结果 [2]，关于其行为的信息很少。 Kiv(z) 当 v 和 z 都很大时，这种情况在应用中非常重要。在[2]中，应用了兰格微分方程方法，并给出了函数及其导数的零点的渐近公式。本文的目的是对 v—» 00 的 Kiv(z) 给出相当完整的描述；证明将在别处给出。基于 (1) 和 Olver 定理 B [4]、[6]，根据 Airy 函数 Ai(%) 及其导数 Ai'ig) 为区域 R 中的 p-*oo 构造一致渐近级数，该区域包含扇区 Re s^O, 35*0。
The modified Bessel equation dw 1 dw ( v\ (1) + ( l U « 0 dz z dz \ z) with its particular solution w**Ki»(z)t the modified Bessel function of the third kind with pure imaginary order is of fundamental significance in the diffraction theory of pulses. Moreover this function is the kernel of the Lebedev transform [3]. With the exception of Friedlander's results [2] little information is available about the behavior of Kiv{z) when both v and z are large which case is of great importance in the applications. In [2] Langer's differential equation method is applied and an asymptotic formula is given for the function and for the zeros of its derivative. The aim of this paper is to give a fairly complete description of Kiv{z) for v—» 00 ; the proofs will be given elsewhere. Based on (1) and on Olver's Theorem B [4], [6] a uniform asymptotic series is constructed in terms of the Airy function Ai(%) and of its derivative Ai'ig) for p-*oo in a region R which contains the sector Re s^O, 35*0.