# §30.2(i) Spheroidal Differential Equation

 30.2.1 $\frac{d}{dz}\left((1-z^{2})\frac{dw}{dz}\right)+\left(\lambda+\gamma^{2}(1-z^{% 2})-\frac{\mu^{2}}{1-z^{2}}\right)w=0.$

This equation has regular singularities at $z=\pm 1$ with exponents $\pm\frac{1}{2}\mu$ and an irregular singularity of rank 1 at $z=\infty$ (if $\gamma\neq 0$). 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.

# §30.2(ii) Other Forms

The Liouville normal form of equation (30.2.1) is

 30.2.2 $\frac{{d}^{2}g}{{dt}^{2}}+\left(\lambda+\frac{1}{4}+\gamma^{2}{\mathop{\sin\/}% \nolimits^{2}}t-\frac{\mu^{2}-\frac{1}{4}}{{\mathop{\sin\/}\nolimits^{2}}t}% \right)g=0,$
 30.2.3 $\displaystyle z$ $\displaystyle=\mathop{\cos\/}\nolimits t,$ $\displaystyle w(z)$ $\displaystyle=(1-z^{2})^{-\frac{1}{4}}g(t).$

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

 30.2.4 $(\zeta^{2}-\gamma^{2})\frac{{d}^{2}w}{{d\zeta}^{2}}+2\zeta\frac{dw}{d\zeta}+% \left(\zeta^{2}-\lambda-\gamma^{2}-\frac{\gamma^{2}\mu^{2}}{\zeta^{2}-\gamma^{% 2}}\right)w=0.$

# §30.2(iii) Special Cases

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).