A NOTE ON DUAL INTEGRAL EQUATIONS INVOLVING ASSOCIATED LEGENDRE FUNCTION

This note is concerned with a formal method to obtain a closed form solution of certain dual integral equations involving the associated Legendre function P−1/2


INTRODUCHON.
Dual integral equations involving the associated Legendre function P_'/(cosh ,) (m -0,1, 2, ...) were considered by Rukhovets and Ufliand [1] who expressed the solution in terms ofone unknown function which satisfies a Fredholm integral equation of second kind.Later Pathak [2] considered several dual I integral equations mvolv]ng P /,(cosh a) where -< Ret < .He exploited the results of some integrals _ involving P/,(cosh a) to handle these dual integral equations and obtained closed form solution in some cases and reduced them to the solution of a Fredholm integral equation of second kind in other cases.
For 0 the corresponding integral equations were mostly considered earlier by Babloian [3].
In the present note we have considered a closed form solution of the following dual integral equations o [ A0:)P ,(cosh ,)d gO,), , > a, ( withRe < .We fit find a cloud fo lution e ir of dual teal uations isting d (1.1)   and (1.2) wi g(a) 0, and en e lution d e ir isting d (1.1) with a) 0 d (1.2 e desir mlution of e dual teal equatio (1.1) and (1.2) en taed by combing lutio of the o pai.both s rul of o tals volving P.z,(cosh a) ve en util to ndle em.
and 1 [ P.,,(cosh .)costdl: where R i" The case I-O, g(a)-0 was considered earlier by Dhaliwal and Singh [4] who obtained a closed form solution of the corresponding dual integral equations.The method used in this note is formal and may be regarded as a generalization of the method given in [4].No attempt has been made to find precise conditions under which the solution obtained here is valid.

SOLUTION WHEN g(a) 0
In this section we find a closed form solution of the following dual integral equations 0<a<a, (2.1) l<l<k.
Its solution can be written in several different forms (cf.Cooke [5]).One such form is then (2.6) reduces to the familiar form where t) m fod out from (Z3) d (2.2 we bstitute (2.5) (2.3) d terchange e er of teation (uming is m valid) we find 1 -c + J0n -F(x), 0 <x <a, 2 which, er differentiation with m x u e sinlu inteal uation for t) ven by m , o < < , C si where e mal e of e uchy wcipal value.we substitute where z-lB(z)tanh czcos xadz F(x), 0 <x <a, (2.3) with Rett < .
3. SOLIYHON WHEN.f(a) = 0 In this section we find a closed form solution of the dual integral equations il:-lc(l:)e_+(cosh a)tanh cv.d: 0, 0 < a < a, fo C('t:)P__ (cosh a)d: O t>a, with Rel < .By using a procedure similar to that of section 2, (3.1) reduces to