A New Method of Evaluation of Howland Integrals

In this paper, two Howland integrals are evaluated to 25D when the index is an odd integer. Those Howland integrals when the index is an even integer have been evaluated to 18D by Nelson. A new method of evaluation is used. The four Howland integrals were first evaluated to 5D by Howland himself, partly with Stevenson, in the papers dealing with a perforated strip [1], [2]. Ling and Nelson in an earlier paper [3] evaluated these integrals to 6D by using a different method through some intermediate integrals. Later, Ling [4] reproduced the 6D values and also added values of a group of related integrals. Recently, Nelson [5], by using the same method, evaluated the integrals to 9D. In the process of computing some related integrals arising from axisymmetrical problems, Nelson, in the same paper, further evaluated the following two Howland integrals to 18D, when k is an even integer: Ik _ 1 f" wkdw ik^l), UJ It 2ik\) Jo sinh w ± w ' ik ^ 3). The aim of the present paper is to evaluate these two integrals to 25D, when k is an odd integer, by using a direct method without recourse to the intermediate integrals. We begin by expanding the integrands into series as follows: k « k —v w 2w e sinh w ± w i ± 2we"° e~2" we 2-, (Tl)/z„(H>)e , n-0 where pn(w) is the Gegenbauer polynomial of degree n and order unity [6]. The expressions are found to be different depending on zz being an even or an odd integer. They are, for zz S: 0, P2 (3) ov)= £i-irn(nt W)(2w)2TM, m-o \ 2m I , -. \-> , .vi+m/ZZ ~T ZZZ + 1 \.. ,2m+l With the aid of the integral (4) 1 wme "" dw = -sil , (a > 0) Jo a Received October 27, 1969, revised April 21, 1970. AMS 1969 subject classifications. Primary 6525, 6505.