In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.

A modular function is a function that, like a modular form, is invariant with respect to the modular group, but without the condition that f (z) be holomorphic in the upper half-plane. Instead, modular functions are meromorphic.

Modular form theory is a special case of the more general theory of automorphic forms, and therefore can now be seen as just the most concrete part of a rich theory of discrete groups.

Source: Modular form – Wikipedia