29.2.1 | |||
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.
Next, let be any real constants that satisfy and
29.2.6 | ||||
(These constants are not unique.) Then with
29.2.7 | ||||
29.2.8 | ||||
we have
29.2.9 | |||
and
29.2.10 | |||
where
29.2.11 | |||
with
29.2.12 | ||||
For the Weierstrass function see §23.2(ii).