InverseWeierstrassP

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

General characteristics

Domain and analyticity

(℘^(-1))(z;(Subscript[g, 2]),(Subscript[g, 3])) is an analytical function  of z, (Subscript[g, 2]) and (Subscript[g, 3]) which is defined in (ℂ^3).

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to (Subscript[g, 3])

The function (℘^(-1))(z;(Subscript[g, 2]),(Subscript[g, 3])) does not have poles and essential singularities with respect to (Subscript[g, 3]).

With respect to (Subscript[g, 2])

The function (℘^(-1))(z;(Subscript[g, 2]),(Subscript[g, 3])) does not have poles and essential singularities with respect to (Subscript[g, 2]).

With respect to z

The function (℘^(-1))(z;(Subscript[g, 2]),(Subscript[g, 3])) does not have poles and essential singularities with respect to z.

Branch points

With respect to (Subscript[g, 3])

With respect to (Subscript[g, 2])

With respect to z

Branch cuts

Branch cut locations: complicated

Series representations

Generalized power series

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

Expansions at z==0

Integral representations

On the real axis

Of the direct function

Differential equations

Ordinary nonlinear differential equations

Identities

Functional identities

Differentiation

Low-order differentiation

Symbolic differentiation

Representations through more general functions

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With inverse function

History

– L. Euler (1761)
– J.-L. Lagrange (1769)

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