ModularLambda

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

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

Symmetries and periodicities

Periodicity

λ(z) is a periodic function with period 2.

Poles and essential singularities

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

Branch points

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

Branch cuts

The function λ(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])

Expansions at z==0

Expansions at z==∞

Product representations

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 related functions

Involving Weierstrass functions

Involving theta functions

Involving other related functions

History

– N .H. Abel
– C. G. J. Jacobi
– C. Hermite

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