In mathematics, a Maass wave form is a function on the upper half plane that transforms like a modular form

Modular form

In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and 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...

 but need not be holomorphic

Holomorphic function

In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

. They were first studied by .

Definition

A Maass wave form is defined to be a continuous complex-valued function f of τ = x + iy in the upper half plane satisfying the following conditions:

• f is invariant under the action of the group SL₂(Z)
  Modular group
  In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...
  
   on the upper half plane.
• f is an eigenvector of the Laplacian operator
• f is of at most polynomial growth at cusps
  Cusp form
  In number theory, a branch of mathematics, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion \Sigma a_n q^n...
  
  of SL₂(Z).

A weak Maass wave form is defined similarly but without the growth condition at cusps.