§30.2 Differential Equations

This equation has regular singularities at $z=\pm 1$ with exponents $\pm \frac{1}{2}\mu$ and an irregular singularity of rank 1 at $z=\mathrm{\infty }$ (if $\gamma \ne 0$). The equation contains three real parameters $\lambda$, ${\gamma }^2$, and $\mu$. In applications involving prolate spheroidal coordinates ${\gamma }^2$ is positive, in applications involving oblate spheroidal coordinates ${\gamma }^2$ is negative; see §§30.13, 30.14.

The Liouville normal form of equation (30.2.1) is

With $\zeta =\gamma z$ Equation (30.2.1) changes to

If $\gamma =0$, Equation (30.2.1) is the associated Legendre differential equation; see (14.2.2). If ${\mu }^2=\frac{1}{4}$, Equation (30.2.2) reduces to the Mathieu equation; see (28.2.1). If $\gamma =0$, Equation (30.2.4) is satisfied by spherical Bessel functions; see (10.47.1).