Let and be periods of a doubly periodic function, with the half-period ratio a number with . Then Klein's absolute invariant (also called Klein's modular function) is defined as

where and are the invariants of the Weierstrass elliptic function with modular discriminant

(Klein 1877). If , where H is the upper half-plane, then

is a function of the ratio only, as are , , and . Furthermore, , , , and are analytic in H (Apostol 1997, p. 15).

Klein's absolute invariant is implemented in Mathematica as KleinInvariantJ[tau].

The function is the same as the j-function, modulo a constant multiplicative factor.

Every rational function of J is a modular function, and every modular function can be expressed as a rational function of J (Apostol 1997, p. 40).

Klein's invariant can be given explicitly by

(Klein 1878/79, Cohn 1994), where is the elliptic lambda function

is a Jacobi theta function, the are Ramanujan-Eisenstein series, and q is the nome. Klein's invariant can also be simply expressed in terms of the five Weber functions , , , , and .

is invariant under a unimodular transformation, so

and is a modular function. takes on the special values

satisfies the functional equations

It satisfies a number of beautiful multiple-argument identities, including the duplication formula

with
and the Dedekind eta function, the triplication formula
with
and the quintuplication formula
with

Plotting the real or imaginary part of in the complex plane produces a beautiful fractal-like structure illustrated above.

 

Elliptic Lambda Function, j-Function, Jacobi Theta Functions, Pi, Ramanujan-Eisenstein Series, Weber Functions