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 harv|Maass|1949.

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 $-y^2left(frac\{partial^2\}\{partial\; x^2\}\; +\; frac\{partial^2\}\{partial\; y^2\}\; ight)$
*"f" is rapidly decreasing at cusps of SL₂(Z).

ee also

*Mock modular form
*Real analytic Eisenstein series

References

*Citation | last1=Bump | first1=Daniel | title=Automorphic forms and representations | publisher=Cambridge University Press | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-55098-7 | id=MathSciNet | id = 1431508 | year=1997 | volume=55
*Citation | last1=Maass | first1=Hans | authorlink=Hans Maass|title=Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen | doi=10.1007/BF01329622 | id=MathSciNet | id = 0031519 | year=1949 | journal=Mathematische Annalen | issn=0025-5831 | volume=121 | pages=141–183