Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











WeierstrassPHalfPeriodValues






Mathematica Notation

Traditional Notation









Elliptic Functions > WeierstrassPHalfPeriodValues[{g2,g3}] > Introduction to the Weierstrass utility functions





The best-known properties and formulas for Weierstrass utilities


The Weierstrass invariants have the following values at infinities:

The Weierstrass function values at half-periods can be evaluated at closed forms for some values of arguments , :

The Weierstrass zeta function values at half-periods can also be evaluated at closed forms for some values of arguments , :

The Weierstrass half‐periods , the Weierstrass function values at half-periods , and the Weierstrass zeta function values at half-periods are vector‐valued functions of and that are analytic in each vector component, and they are defined over .

The Weierstrass invariants is a vector‐valued function of and that is analytic in each vector component, and it is defined over (for ).

The Weierstrass invariants with is a periodic function with period :

The other Weierstrass utility functions , , and are not periodic functions.

The Weierstrass half‐periods and Weierstrass zeta function values at half-periods have mirror symmetry:

The Weierstrass invariants and the Weierstrass function values at half-periods have standard mirror symmetry:

The Weierstrass invariants have permutation symmetry and are homogeneous:

The Weierstrass invariants are the invariants under the change of variables and with integers , , , and , satisfying the restriction (modular transformations):

This property leads to similar properties of the Weierstrass function values at half-periods and the Weierstrass zeta function values at half-periods :

The Weierstrass half‐periods and invariants have the following double series expansions:

where is a Klein invariant modular function.

The last double series can be rewritten in the following forms:

The Weierstrass invariants , the Weierstrass function values at half-periods , and the Weierstrass zeta function values at half-periods have numerous q‐series representations, for example:

where .

The following rational function of and is a modular function if considered as a function of :

The Weierstrass utilities have some other forms of series expansions, for example:

where is the divisor sigma function.

The Weierstrass half‐periods and invariants have the following integral representations:

The Weierstrass utilities can have product representations. For example, the Weierstrass function values at half-periods can be expressed through the following products:

where .

The Weierstrass utilities satisfy numerous identities, for example:

The first derivatives of Weierstrass half‐periods and the Weierstrass and zeta function values at half-periods and with respect to variable and have the following representations:

where are the values of the derivative of the Weierstrass elliptic function at half-period points .

The first derivatives of Weierstrass invariants with respect to the variables and can be represented in different forms:

The -order derivatives of Weierstrass invariants with respect to the variables and have the following representations:

The indefinite integrals of Weierstrass invariants with respect to the variable have the following representations:

The Weierstrass half‐periods satisfy the following differential equations:

The Weierstrass invariants satisfy the following differential equations:

The Weierstrass zeta function values at half-periods satisfy the following differential equations:





© 1998-2014 Wolfram Research, Inc.