# §30.2 Differential Equations

## §30.2(i) Spheroidal Differential Equation

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

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 $\frac{{\mathrm{d}}^{2}g}{{\mathrm{d}t}^{2}}+\left(\lambda+\frac{1}{4}+\gamma^{% 2}{\sin}^{2}t-\frac{\mu^{2}-\frac{1}{4}}{{\sin}^{2}t}\right)g=0,$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $\sin\NVar{z}$: sine function, $\lambda$: real parameter, $\gamma^{2}$: real parameter and $\mu$: real parameter Referenced by: §30.2(iii) Permalink: http://dlmf.nist.gov/30.2.E2 Encodings: TeX, pMML, png See also: Annotations for §30.2(ii), §30.2 and Ch.30
 30.2.3 $\displaystyle z$ $\displaystyle=\cos t,$ $\displaystyle w(z)$ $\displaystyle=(1-z^{2})^{-\frac{1}{4}}g(t).$ ⓘ Symbols: $\cos\NVar{z}$: cosine function, $z$: complex variable and $w$: solution to DE Permalink: http://dlmf.nist.gov/30.2.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §30.2(ii), §30.2 and Ch.30

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

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

## §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).