Asymptotic Evaluation of Integrals Involving a Fractional Derivative

Abstract

For the integral \[\int_\alpha ^\infty {e^{ - z(t - a )} I^{\lambda - 1} f(t)dt} \] an asymptotic expansion is obtained as $z \to \infty $. Here $\lambda $ is fixed, $0 < \lambda < 1, $, $I^{\lambda - 1} $ is the operator of fractional integration, and the expansion holds uniformly for $a \geqq 0$. A similar expansion is obtained for the integral from 0 to a and is applied to the solution of an integral equation.