Modular functions are special types of functions in mathematics that are defined on the complex numbers and exhibit periodic behavior. They are often studied in the context of number theory and have applications in various areas, including cryptography and string theory. A key feature of modular functions is their transformation properties under the action of modular transformations, which relate to the symmetries of the complex upper half-plane. One of the most famous examples of a modular function is the j-invariant, which classifies elliptic curves over the complex numbers. Modular functions can also be connected to modular forms, which are similar but have additional properties, such as being holomorphic and having specific growth conditions. Together, these concepts play a crucial role in modern mathematics and theoretical physics.