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
, respectively, in terms of non-oscillating integrals. Starting from two well-known integral representations of
[see G. N. Watson
, A treatise on the theory of Bessel functions (1944); p. 181, §6.22 (3) and (7)] in which
are assumed to be real,
, and making use of certain paths of steepest descent (the saddle point contours), the author deduces interesting non-oscillating integral representations for
. These representations can be useful for obtaining asymptotic expansions as well as numerical algorithms.