§23.6 Relations to Other Functions

§23.6(i) Theta Functions

In this subsection , are any pair of generators of the lattice , and the lattice roots , , are given by (23.3.9).

23.6.1
23.6.2
23.6.3
23.6.4
23.6.8

With ,

For further results for the -function see Lawden (1989, §6.2).

§23.6(ii) Jacobian Elliptic Functions

Again, in Equations (23.6.16)–(23.6.26), are any pair of generators of the lattice and are given by (23.3.9).

23.6.16

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).

§23.6(iii) General Elliptic Functions

For representations of general elliptic functions (§23.2(iii)) in terms of and see Lawden (1989, §§8.9, 8.10), and for expansions in terms of see Lawden (1989, §8.11).

§23.6(iv) Elliptic Integrals

¶ Rectangular Lattice

Let be on the perimeter of the rectangle with vertices . Then is real (§§23.5(i)23.5(ii)), and

For (23.6.30)–(23.6.35) and further identities see Lawden (1989, §6.12).