# §23.15 Definitions

## §23.15(i) General Modular Functions

In §§23.1523.19, and denote the Jacobi modulus and complementary modulus, respectively, and () denotes the nome; compare §§20.1 and 22.1. Thus

Also denotes a bilinear transformation on , given by

in which are integers, with

23.15.4

The set of all bilinear transformations of this form is denoted by SL (Serre (1973, p. 77)).

A modular function is a function of that is meromorphic in the half-plane , and has the property that for all , or for all belonging to a subgroup of SL,

where is a constant depending only on , and (the level) is an integer or half an odd integer. (Some references refer to as the level). If, as a function of , is analytic at , then is called a modular form. If, in addition, as , then is called a cusp form.

## §23.15(ii) Functions , ,

### ¶ Elliptic Modular Function

23.15.6

compare also (23.15.2).

### ¶ Klein’s Complete Invariant

23.15.7

where (as in §20.2(i))

### ¶ Dedekind’s Eta Function (or Dedekind Modular Function)

In (23.15.9) the branch of the cube root is chosen to agree with the second equality; in particular, when lies on the positive imaginary axis the cube root is real and positive.