KleinInvariantJ

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

Values at complex z

Values at quadratic irrationalities (Heegner Numbers)

Values at infinities

General characteristics

Domain and analyticity

J(z) is an analytical function of z which is defined over the upper half of the complex z-plane.

Symmetries and periodicities

Periodicity

Poles and essential singularities

On the boundary of analyticity the function J(z) has a dense set of poles.

Branch points

The function J(z) does not have branch points.

Branch cuts

The function J(z) does not have branch cuts.

Natural boundary of analyticity

The real axis Im(z)==0 is the natural boundary of the region of analyticity.

Series representations

Generalized power series

Expansions at generic point z==(Subscript[z, 0])

Exponential Fourier series

Differential equations

Ordinary nonlinear differential equations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Identities

Functional identities

Differentiation

Low-order differentiation

Operations

Limit operation

Representations through equivalent functions

With inverse function

With related functions

Involving Weierstrass functions

Involving theta functions

Involving other related functions

Zeros

Theorems

Property of modular functions

Every modular function m(τ) (meaning a function that is invarant under all argument substitutions of the form z->(a z + b)/(c z + d) where a, b, c, and d are integers and a d-b c==1) is a rational function of J(z) .

Transcendentality of Klein invariant J for algebraic argument

The value J(α) is transcendental for any algebraic α where Im(α)>0 and α not being a quadratic irrational.

Moonshine conjecture

The dimensions of the representations of the monster group (the largest simple sporadic group) of order (2^46)×(3^20)×(5^9)×(11^2)×(13^3)×17 ×19× 23×29×31×41×47×59×71 are simple combinations the Fourier coefficients of 1728 J(z).

Solution of Chazy equation

The function w(z)==(((4-7 J(τ)) ((J'(τ))^2)+6 (J(τ)-1) J(τ) J''(τ))/(2 (J(τ)-1) J(τ) J'(τ))) is a solution of Chazy equation w'''(z)==2 w''(z) w(z)-3 ((w(z))^2).

Replicability of 1728 J(τ)-744

The function 1728 J(τ)-744 is completely replicable:

J(σ)- J(τ)==((e^(-2π i σ))/1728)(∏_(n=-1)^∞)(∏_(m=1)^∞)(((1-(e^(2π i( m σ+n τ )))))^(c(n m)))/;1728 J(τ)-744==(∑_(n=-1)^∞)c(n)(e^(2π i τ)).

History

– R. Dedekind (1877)
– F. Klein (1878)
– R. Fricke (1890–1892)

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