Evaluation of Integrals of Howland Type Involving a Bessel Function

This paper presents a method of evaluation of four integrals of Howland type, which involve a Bessel function in the integrands. With the aid of tabulated values, they are evaluated to 10D. Two of the four Howland integrals needed in the evaluation are evaluated anew to 20D in order to provide adequate accuracy. In a recent investigation of certain problems in elasticity concerning elliptic boundaries, four integrals of Howland type involving an additional Bessel function in the integrands were encountered. We believe that they deserve special consideration. The integrals are as follows: (1) F"'k( \ = tC mkj^ma"> d (n + k> I), FZk{a) k\ J0 sinh 2m ± 2m (n + k > 3), E"'k( \ = — (°° mkjn(ma) coth m dm (n + k > 2), E*,k k\J0 sinh 2m ± 2m m (n + k > 4), where Jn is a Bessel function of the first kind of integral order n. n and k are nonnegative integers restricted as indicated above in order to render each integral convergent at the lower limit. The constant a may be real or complex. By using the usual series expression for Jn and integrating, the first integral