# §29.2 Differential Equations

## §29.2(i) Lamé’s Equation

where and are real parameters such that and . For see §22.2. This equation has regular singularities at the points , where , and , are the complete elliptic integrals of the first kind with moduli , , respectively; see §19.2(ii). In general, at each singularity each solution of (29.2.1) has a branch point (§2.7(i)). See Figure 29.2.1.

## §29.2(ii) Other Forms

Next, let be any real constants that satisfy and

29.2.6

(These constants are not unique.) Then with

we have

and

29.2.10

where

with

29.2.12

For the Weierstrass function see §23.2(ii).

Equation (29.2.10) is a special case of Heun’s equation (31.2.1).