Spheroidal Functions of the Second Kind
Julius Adams Stratton · Proceedings of the National Academy of Sciences · 1935
In a recent note in these PROCEEDINGS,I henceforth to be referred to as I, certain solutions of the equation (1-z2)w'-2(a +1)zw' + (b-c2z2)w=° (1) were defined.Corresponding to a discrete set of characteristic values of the parameter b1 there exist solutions of the type Re ,(c, z) = (CZ) -a-1/2 E al Jx+a+1/2(CZ)t (I-10) n convergent throughout the entire z-plane and behaving as (cz)-a-i sin cz-+ ar as z-* co.In the neighborhood of z = 0 a formal solution may be obtained in terms of an expansion in hypergeometric functions, 54,(c, z) -' d, Tx(), (I-8) which for the characteristic values b1 is also bounded at the singularities z = k 1.With the aid of the integral representation (I-18) it was shown that (I-8) and (I-10) are proportional,