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

Dual integral equations of the type \[ \int _0^\infty {{u^\lambda }f\left ( u \right ){J_\mu }\left ( {ru} \right )du = g\left ( r \right ),0 < r < 1, \\ \int _0^\infty u {{\left ( {{u^2} + {a^2}} \right )}^{ - 1/2}}f\left ( u \right ){J_v}\left ( {ru} \right )du = h\left ( r \right ),} 1 < r < \infty ,\] where $g\left ( r \right )$, $h\left ( r \right )$ are prescribed functions and $f\left ( u \right )$ 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.