In this paper are investigated some mathematical properties of spheroidal functions satisfying the differential equation of the form:
\frac
ddz\left{(1−
z2)\frac
dZdz\
ight}+(λ+κ
2z2)
Z=0.
One solution pe
n(
z) regular at |
z|=1, which has already been known as special case of the generalized spheroidal function pe
nm(
z), is developed into a Legendre expansion and its coefficients are obtained explicitly for several cases.
The solution of the first kind Re
n(
z) and that of the second kind Se
n(
z), which are valid especially when
z>>1, are defined by the definite integrals and are also expressed in series forms in terms of the modified Bessel functions.
An alternative expression for Se
n(
z), which is conveniently used even when
z is not so large, is also defined in like manner as in the case of the derivation of the modified Mathieu function FEK
n(
z) or GEK
n(
z). Further, the asymptotic behaviours of these functions Re
n(
z) and Se
n(
z) are obtained.
Detailed calculations are developed for the case of a prolate spheroid. For the case of an oblate spheroid some essential parts only are given In Appendix.
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