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Zbl 0848.33003
Temme, N.M.
Steepest descent paths for integrals defining the modified Bessel functions of imaginary order. (English)
[J] Methods Appl. Anal. 1, No.1, 14-24 (1994). ISSN 1073-2772

The aim of this paper is to derive representations of the modified Bessel functions of the first and the third kind of purely imaginary orders $K_{i\nu} (x)$ and $I_{i\nu} (x)$, respectively, in terms of non-oscillating integrals. Starting from two well-known integral representations of $K_{i\nu} (x)$ and $I_{i\nu} (x)$ [see {\it G. N. Watson}, A treatise on the theory of Bessel functions (1944); p. 181, \S 6.22 (3) and (7)] in which $\nu$ and $x$ are assumed to be real, $x>0$, $\nu\geq 0$, and making use of certain paths of steepest descent (the saddle point contours), the author deduces interesting non-oscillating integral representations for $K_{i\nu} (x)$ and $I_{i\nu} (x)$. These representations can be useful for obtaining asymptotic expansions as well as numerical algorithms.
[N.Hayek (La Laguna)]

MSC 2000:: *33C10 Cylinder functions, etc.

Keywords: modified Bessel functions of purely imaginary order; paths of steepest descent; saddle point contours

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