In mathematics, Maass forms or Maass wave forms are studied in the theory of automorphic forms. Maass forms are complex-valued smooth functions of the upper half plane, which transform in a similar way under the operation of a discrete subgroup Γ {\displaystyle \Gamma } of S L 2 ( R ) {\displaystyle \mathrm {SL} _{2}(\mathbb {R} )} as modular forms. They are eigenforms of the hyperbolic Laplace operator Δ {\displaystyle \Delta } defined on the upper half plane and satisfy certain growth conditions at the cusps of a fundamental domain of Γ {\displaystyle \Gamma } . In contrast to modular forms, Maass forms need not be holomorphic. They were studied first by Hans Maass in 1949.
General remarks The group
G := S L 2 ( R ) = { ( a b c d ) ∈ M 2 ( R ) : a d − b c = 1 } {\displaystyle G:=\mathrm {SL} _{2}(\mathbb {R} )=\left\{{\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}\in M_{2}(\mathbb {R} ):ad-bc=1\right\}}
operates on the upper half plane
H = { z ∈ C : Im ( z ) > 0 } {\displaystyle {\mathcal {H}}=\{z\in \mathbb {C} :\operatorname {Im} (z)>0\}}
by fractional linear transformations:
( a b c d ) ⋅ z := a z + b c z + d . {\displaystyle {\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}\cdot z:={\frac {az+b}{cz+d}}.}
It can be extended to an operation on H ∪ { ∞ } ∪ R {\displaystyle {\mathcal {H}}\cup \{\infty \}\cup \mathbb {\mathbb {R} } } by defining:
( a b c d ) ⋅ z := { a z + b c z + d if c z + d ≠ 0 , ∞ if c z + d = 0 , {\displaystyle {\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}\cdot z:={\begin{cases}{\frac {az+b}{cz+d}}&{\text{if }}cz+d\neq 0,\\\infty &{\text{if }}cz+d=0,\end{cases}}}
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