Evaluation of Distributions Useful in Kontorovich-Lebedev Transform Theory

Abstract

The transform methods of Kontorovich and Lebedev seem to offer a direct approach to the solution of many physical problems involving the geometry of angular sectors. This promise has often been thwarted in practice by the coupling of boundary conditions which makes a direct inverse transform impossible. In the present work, we offer a method to meet this difficulty, based on the evaluation of the distributions listed below in terms of tabulated functions: \[ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow 0} \int_0^\infty {K_{i\nu } (\lambda r)K_{i\xi } (r)\frac{{dr}} {{r^{1 - \varepsilon } }}} , \hfill \\ \mathop {\lim }\limits_{\varepsilon \searrow 0} \int_0^\infty {K_{i\nu } (\lambda r)r^{2m - 1 - i\xi + \varepsilon } dr} , \hfill \\ \mathop {\lim }\limits_{\varepsilon \searrow 0} \int_0^\infty {K_{i\xi } (r)I_{i\nu } (\lambda r)\frac{{dr}} {{r^{1 - \varepsilon } }}} . \hfill \\ \end{gathered} \]