Weierstrass Elliptic Functions are complex functions that arise in the study of elliptic curves and are essential in various areas of mathematics, including number theory and algebraic geometry. They are defined using a lattice in the complex plane, which allows them to exhibit periodic behavior in two directions, making them doubly periodic. These functions can be expressed in terms of a series expansion and have important properties, such as being meromorphic with poles of order two. The Weierstrass ℘-function, a primary example, is used to construct elliptic curves, which are crucial for understanding solutions to polynomial equations and have applications in cryptography and complex analysis.