# §30.2 Differential Equations

## §30.2(i) Spheroidal Differential Equation

This equation has regular singularities at with exponents and an irregular singularity of rank 1 at (if ). The equation contains three real parameters , , and . In applications involving prolate spheroidal coordinates is positive, in applications involving oblate spheroidal coordinates is negative; see §§30.13, 30.14.

## §30.2(ii) Other Forms

The Liouville normal form of equation (30.2.1) is

With Equation (30.2.1) changes to

## §30.2(iii) Special Cases

If , Equation (30.2.1) is the associated Legendre differential equation; see (14.2.2). If , Equation (30.2.2) reduces to the Mathieu equation; see (28.2.1). If , Equation (30.2.4) is satisfied by spherical Bessel functions; see (10.47.1).