EllipticNomeQ

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

General characteristics

Domain and analyticity

q(m) is an analytical function of m which is defined over the whole complex m-plane.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

The function q(m) does not have poles and essential singularities.

Branch points

Branch cuts

Series representations

Generalized power series

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

Expansions at m ==0

Expansions at m==1

Expansions at m==∞

Differential equations

Ordinary nonlinear differential equations

Differentiation

Low-order differentiation

Symbolic differentiation

Summation

Infinite summation

Operations

Limit operation

Representations through more general functions

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With inverse function

History

– C. G. J. Jacobi (1828) for real arguments
– K. Weierstrass (1883) for complex arguments

Applications include q -series for elliptic functions.

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