Modular form

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In number theory and complex analysis, a modular form is a type of function of a complex number variable that possesses a high degree of symmetry, of a certain kind. Similarly to a periodic function of a real variable, a modular form repeats or transforms in a certain way when its argument is subjected to a particular transformation. Unlike an ordinary periodic function, its symmetries include transformations such as replacing a complex number z by −1/z, and the transformation law is not an exact symmetry of the function, but more like the transformation law of a quasiperiodic function: the function picks up an additional factor, depending on the transformation. Modular forms serve as an important bridge between complex analysis, number theory, and geometry. Modular forms also appear in other areas, such as algebraic topology, sphere packing, and string theory. More precisely, a modular form is a holomorphic function on the complex upper half-plane that roughly satisfies a functional equation...

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