# Maass wave form

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In mathematics, a Maass wave form or Maass form is a function on the upper half plane that transforms like a modular form but need not be holomorphic. They were first studied by Hans Maass in Maass (1949).

## Definition

Let k be a half-integer and s be a complex number. A Maass form of weight k for Γ with Laplace eigenvalue s is a smooth function from the upper half-plane to the complex numbers satisfying the following conditions:

• For all $\gamma = \left(\begin{smallmatrix} a & b \\ c & d\end{smallmatrix}\right) \in \Gamma$ and all $\tau \in \mathbb{H}$, we have $f\left(\frac{a\tau+b}{c\tau+d}\right) = (c\tau+d)^k f(\tau)$.
• We have $\Delta_{k} f = s f$, where $\Delta_{k}$ is the weight k hyperbolic laplacian defined as $\Delta_{k} = -y^{2} \left(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}\right) i k y \left(\frac{\partial}{\partial x} + i \frac{\partial}{\partial y}\right)$.
• The function f is of at most polynomial growth at cusps.

A weak Maass wave form is defined similarly but with the third condition replaced by "The function f has at most linear exponential growth at cusps". Moreover, f is said to be harmonic if it is annihilated by the Laplacian operator.