TeXipedia

weierp

Represents the Weierstrass elliptic function, commonly used in complex analysis and number theory.

Overview

A fundamental mathematical function that plays a key role in the study of elliptic functions and complex analysis.

  • Essential in the theory of doubly periodic functions
  • Frequently appears in advanced mathematics dealing with elliptic curves
  • Used to solve certain nonlinear differential equations
  • Important in mathematical physics, particularly in the study of periodic phenomena

Examples

Using Weierstrass p-function in an elliptic function equation.

℘(z;ω1,ω2)=1z2+∑ω(1(z−ω)2−1ω2)\weierp(z; \omega_1, \omega_2) = \frac{1}{z^2} + \sum_{\omega} \left(\frac{1}{(z-\omega)^2} - \frac{1}{\omega^2}\right) 
\weierp(z; \omega_1, \omega_2) = \frac{1}{z^2} + \sum_{\omega} \left(\frac{1}{(z-\omega)^2} - \frac{1}{\omega^2}\right)

Defining a complex torus using the Weierstrass p-function.

y2=4(x−e1)(x−e2)(x−e3),x=℘(z)y^2 = 4(x-e_1)(x-e_2)(x-e_3), \quad x = \weierp(z) 
y^2 = 4(x-e_1)(x-e_2)(x-e_3), \quad x = \weierp(z)

Expressing the derivative of the Weierstrass p-function.

℘′(z)2=4℘(z)3−g2℘(z)−g3\weierp'(z)^2 = 4\weierp(z)^3 - g_2\weierp(z) - g_3 
\weierp'(z)^2 = 4\weierp(z)^3 - g_2\weierp(z) - g_3