The paper deals with for large values of . Following Olver, the author transforms the differential equation such that solutions suitable for asymptotic investigation emerge. Auxiliary variables , , are introduced, and is taken as the large expansion parameter. As a result of rather long calculations, including a discussion of error bounds, the following result is obtained. If the -plane is cut from to , and is real, then
where denotes a modified Bessel function, and the coefficient functions and satisfy certain differential-recursion equations. The author’s expansion has a wider range of validity than an expansion in powers of given by Watson. Further results based upon transformations of are considered. Also, results involving Legendre functions are noted as particular cases. Finally, the author derives an expansion where is replaced with . The coefficient functions again satisfy a differential-recursion equation.