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# Maass wave form

In mathematics, a Maass wave 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 Maass .

## 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) on the upper half plane.
• f is an eigenvector of the Laplacian operator $-y^2left\left(frac\left\{partial^2\right\}\left\{partial x^2\right\} + frac\left\{partial^2\right\}\left\{partial y^2\right\}right\right)$
• f is rapidly decreasing at cusps of SL2(Z).