A Pair of Dual Integral Equations Occurring in Diffraction Theory
Proceedings of the Edinburgh Mathematical Society
Dual integral equations of the form f 00f Jo where f(x) and g(x) are given functions, \j/(x) is unknown, k^.0, \i, v and a are real constants, have applications to diffraction theory and also to dynamical problems in elasticity. The special cases v = -\i, a = 0 and v = n = 0, 0 < a 2 < l were treated by Ahiezer (1). More recently, equations equivalent to the above were solved by Peters (2) who adapted a method used earlier by Gordon (3) for treating the (extensively studied) case /x = v, k = 0.
... case /x = v, k = 0. We present here a method of solution which has affinities with the " elementary " method introduced by Sneddon (4) for the case n = v = k = 0 and developed by Copson (5) for the case n = v, k = 0. The essence of the method is the reduction of the dual integral equations to a single integral equation: the integral equations which arise in the study of equations (1) and (2) (see Lemmas 2 and 3 below) have not apparently been treated before and seem to be of interest in their own right; they generalise the integral equations discussed by Copson (6) and Jones (7) . The analysis throughout is formal: orders of integration, and of integration and differentiation, are inverted freely. The solutions obtained may always be verified by substitution into the original dual integral equations. 2. We first state some preliminary results, in the form of three lemmas. Lemma 1. (Sonine's " second " integral, (8), p. 415). f Jo (0, 0Re(ji)> -1.