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| 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 function|holomorphic]]. They were first studied by [[Hans Maass]] in {{harvtxt|Maass|1949}}.
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| ==Definition==
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| 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:
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| *''f'' is invariant under the action of the group [[Modular group|SL<sub>2</sub>('''Z''')]] on the upper half plane.
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| *''f'' is an [[eigenvector]] of the [[Laplacian operator]] <math>-y^2\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right).</math>
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| *''f'' is of at most polynomial growth at [[Cusp form|cusps]] of SL<sub>2</sub>('''Z''').
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| A '''weak Maass wave form''' is defined similarly but without the growth condition at cusps.
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| ==See also==
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| *[[Mock modular form]]
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| *[[Real analytic Eisenstein series]]
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| ==References==
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| *{{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}}
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| *{{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]] | volume=121 | pages=141–183}}
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| [[Category:Modular forms]]
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Latest revision as of 00:27, 29 July 2014
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