Bessel Functions of Purely Imaginary Order, with an Application to Second-Order Linear Differential Equations Having a Large Parameter

Bessel functions of purely imaginary order are examined. Solutions of both the modified and unmodified Bessel equations are defined which, when their order is purely imaginary and their argument is real and positive, are pairs of real numerically satisfactory functions. Recurrence relations, analytic continuation formulas, power series representations, Wronskian relations, integral representations, behavior at singularities, and asymptotic forms of the zeros are derived for these numerically satisfactory functions. Also, asymptotic expansions in terms of elementary and Airy functions are derived for the Bessel functions when their order is purely imaginary and of large absolute value. Second-order linear ordinary differential equations having a large parameter and a simple pole are then examined, for the case where the exponent of the pole is complex. Asymptotic expansions are derived for the solutions in terms of the numerically satisfactory Bessel functions of purely imaginary order.