On the Zeros of a Class of Bessel Functions Whose Argument and Order are Functions of a Complex Variable

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The zeros of a class of modified Bessel functions I (u) and K (u) whose order and argument are analytic functions of the complex variables s are investigated. This is a particular case of more general problems arising when the inverse Laplace transforms of functions describing physical processes are calculated. It is found that the function I (u) has zeros only on the real negative axis in the s-plane, while K (u) can have an infinite set of complex conjugate pairs of zeros in the half-plane Re s<0 together with a finite number of real negative zeros