Maass wave form

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 ${\displaystyle \Gamma }$ of ${\displaystyle \mathrm {SL} _{2}(\mathbb {R} )}$ as modular forms. They are eigenforms of the hyperbolic Laplace operator ${\displaystyle \Delta }$ defined on ${\displaystyle \mathbb {H} }$ and satisfy certain growth conditions at the cusps of a fundamental domain of ${\displaystyle \Gamma }$. In contrast to modular forms, Maass forms need not be holomorphic. They were studied first by Hans Maass in 1949.