Convergent and asymptotic expansions of the displacement elastodynamic integral in terms of known functions

Abstract

The integral ${\int }_0^{\infty }\frac{J_{\mu }\left(rt\right)J_{\nu }\left(Rt\right)}{t^{\alpha }(t-s)}dt$ plays an essential role in the study of several phenomena in the theory of elastodynamics (Ceballos and Prato, 2014). But an exact evaluation of this integral in terms of known functions is not possible. In this paper, we derive an analytic representation of this integral in the form of convergent series of elementary functions and hypergeometric functions. This series have an asymptotic character for either, small values of the variable $s$, or for small values of the variables $r$ and $R$. It is derived by using the asymptotic technique designed in Lopez (2008) for Mellin convolution integrals. Some numerical experiments show the accuracy of the approximation supplied by the first few terms of the expansion.