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Steepest descent paths for integrals defining the modified Bessel functions of imaginary order. (English) Zbl 0848.33003
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, § 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.

33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$