Spheroidal wave function

Remarks

The eigenvalue problem is a singular Sturm–Liouville problem on $(-1, 1)$, which explains the discrete increasing sequence of eigenvalues. Example: for $c = 0$ the equation is the associated Legendre equation, $\lambda_{mn}(0) = n(n + 1)$, and $S_{mn}(0, x)$ is a multiple of the associated Legendre polynomial $P_n^m(x)$. For $c = 0$, $m = 0$ and $n = 0$: $y = 1$ is bounded and satisfies $-2x \cdot 0 + (0 - 0) \cdot 1 = 0$ with $\lambda = 0$. Non-example: for $c = 0$, $m = 0$ and $\lambda = 0$, the function $y = \frac12 \log \frac{1 + x}{1 - x}$ also solves the equation, but it is unbounded near $x = \pm 1$, so it is not a spheroidal wave function. The radial spheroidal wave functions solve the same equation on $x > 1$ and describe the dependence on the other coordinate. Normalisations differ between authors; a common one makes $S_{mn}(c, x)$ tend to $P_n^m(x)$ as $c \to 0$.