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.
For see §22.16(i).
Next, let be any real constants that satisfy and
(These constants are not unique.) Then with
For the Weierstrass function see §23.2(ii).