In this subsection , are any pair of generators of the lattice , and the lattice roots , , are given by (23.3.9).
For further results for the -function see Lawden (1989, §6.2).
Similar results for the other nine Jacobi functions can be constructed with the aid of the transformations given by Table 22.4.3.
For representations of the Jacobi functions , , and as quotients of -functions see Lawden (1989, §§6.2, 6.3).
For relations to symmetric elliptic integrals see §19.25(vi).
Let be a point of different from , and define by
where the integral is taken along any path from to that does not pass through any of . Then , where the value of depends on the choice of path and determination of the square root; see McKean and Moll (1999, pp. 87–88 and §2.5).