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Represents the Weierstrass p-function, a fundamental elliptic function in complex analysis and number theory.

Overview

A crucial mathematical symbol in complex analysis and algebraic geometry, particularly important when working with elliptic functions and curves.

  • Essential in the study of doubly periodic functions and lattices in the complex plane.
  • Frequently appears in advanced number theory problems and research papers.
  • Used to parameterize elliptic curves and solve related differential equations.
  • Common in mathematical physics, especially in the study of periodic phenomena and special functions.

Examples

Using the Weierstrass p-function in an elliptic function equation.

℘(z+ω)=℘(z)\wp(z+\omega) = \wp(z) 
\wp(z+\omega) = \wp(z)

Defining a relation using the Weierstrass p-function.

y2=4℘(z)3−g2℘(z)−g3y^2 = 4\wp(z)^3 - g_2\wp(z) - g_3 
y^2 = 4\wp(z)^3 - g_2\wp(z) - g_3

Expressing the derivative of the Weierstrass p-function.

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