Theta functions are special mathematical functions that arise in various areas of mathematics, particularly in complex analysis and number theory. They are defined using a series expansion and are closely related to elliptic functions. Theta functions can be used to study the properties of lattices and modular forms, making them important in the theory of elliptic curves and modular forms. These functions have applications in physics, particularly in statistical mechanics and string theory. They help describe the behavior of systems at different temperatures and can be used to solve problems involving partition functions. Additionally, theta functions play a role in the study of Riemann surfaces and can be used to construct solutions to certain types of differential equations.