InverseWeierstrassP4

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

(℘^(-1))((Subscript[z, 1]),(Subscript[z, 2]);(Subscript[g, 2]),(Subscript[g, 3])) is the unique value of u for which (Subscript[z, 1])=℘(u;(Subscript[g, 2]),(Subscript[g, 3])) and (Subscript[z, 2])=℘'(u;(Subscript[g, 2]),(Subscript[g, 3])). For (℘^(-1))((Subscript[z, 1]),(Subscript[z, 2]);(Subscript[g, 2]),(Subscript[g, 3])) to exist, (Subscript[z, 1]) and (Subscript[z, 2]) must be related by (Subscript[z, 2]^2)=4 (Subscript[z, 1]^3)-(Subscript[g, 2]) (Subscript[z, 1])-(Subscript[g, 3]).

General characteristics

Domain and analyticity

(℘^(-1))((Subscript[z, 1]),(Subscript[z, 2]);(Subscript[g, 2]),(Subscript[g, 3])) is an analytical function of (Subscript[z, 1]), (Subscript[z, 2]), (Subscript[g, 2]), (Subscript[g, 3]) which is defined in (ℂ^4).

Symmetries and periodicities

Symmetry

No symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to (Subscript[g, 3])

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

With respect to (Subscript[z, 1])

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

Branch cuts

Branch cut locations: complicated

Integral representations

On the real axis

Of the direct function

Differential equations

Ordinary nonlinear differential equations

Differentiation

Low-order differentiation

Symbolic differentiation

Representations through equivalent functions

With inverse function

History

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

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