Jacobi Theta Function

The Jacobi theta function is a special function of two variables defined by an infinite sum of exponential terms, and it is one of a family of four closely related theta functions that together form the foundation of the classical theory of elliptic functions. The functions are named after the German mathematician Carl Gustav Jacob Jacobi, who developed their theory in detail in his 1829 treatise Fundamenta Nova Theoriae Functionum Ellipticarum, though closely related series had already been studied earlier by Leonhard Euler and Carl Friedrich Gauss. Jacobi theta functions satisfy the heat equation in one spatial dimension, provide efficient solutions to problems in number theory such as counting the number of ways an integer can be written as a sum of a fixed number of squares, and are central to the theory of modular forms, where they transform in a precise way under the modular group. Because of these connections, theta functions also reappear in mathematical physics, including in the study of string theory and statistical mechanics.