On Evaluation of Moments of K ν (t)/I ν (t)

Abstract

This paper presents a method of evaluation of the moments of K,(t)/l,(t).iTwo pairs of expressions, each consisting of two series, are obtained according to the index being an even or an odd integer. The method is an extension of the method used by Watson. Values are tabulated to 12D for p = 0(1)2. In a recent paper [1], Roberts considered the computation of the integral (1 ) f dtk J dt (k _ 2v), where I, and K, are modified Bessel functions. He developed the integral into an 'asymptotic series' suitable for computation when k is a large integer. When k is otherwise a small integer, he integrated numerically the following equivalent integral by using Simpson's rule: (2) = k411i(t)dt (k > 2v). In particular, values are tabulated to 8S for v = 1 and k = 2(1)100. Some time ago, Watson [2] evaluated the integral in (2) when k is an even integer. He developed the integral into two series by employing a method based on a modification of Plana's summation formula. It is found that this method can be extended to the case when k is an odd integer. Furthermore, it is also found that the same integral can be developed into different series by a modification of the method. Altogether, two pairs of expressions are obtained according to k being an even or an odd integer. It is the purpose of this paper to present such results. Values of the integral are thereby evaluated to 12D for v' = 0(1)2. It is mentioned that a method analogous to the present one was used recently by the authors for the evaluation of two Howland integrals [3]. For convenience, the integral is redefined, together with a factor, as follows: k+1 (3) ~~~~~~2 kK,(t)_ 7r(k!) lo I,(t) ( v so that it tends asymptotically to unity as k tends to infinity. The equivalent integral is 2k+l1 C tk (4) L 1)! f kdt (k > 2v). 7r(k + 1)! 1I(V The following results are obtained for k > v: Received September 22, 1971. AMS 1969 subject classifications. Primary 6525.