Function-theoretic solution to a class of dual integral equations and an application to diffraction theory

Dual integral equations of the type \[ ∫ 0 ∞ u λ f ( u ) J μ ( r u ) d u = g ( r ) , 0 > r > 1 , ∫ 0 ∞ u ( u 2 + a 2 ) − 1 / 2 f ( u ) J v ( r u ) d u = h ( r ) , 1 > r > ∞ , \int _0^\infty {{u^\lambda }f\left ( u \right ){J_\mu }\left ( {ru} \right )du = g\left ( r \right ),0 > r > 1, \\ \int _0^\infty u {{\left ( {{u^2} + {a^2}} \right )}^{ - 1/2}}f\left ( u \right ){J_v}\left ( {ru} \right )du = h\left ( r \right ),} 1 > r > \infty , \] where g ( r ) g\left ( r \right ) , h ( r ) h\left ( r \right ) are prescribed functions and f ( u ) f\left ( u \right ) is to be found, are solved exactly by the application of function-theoretic methods. As an example, a closed-form solution is obtained for the diffraction of an electromagnetic wave by a plane slit.