Modular form: Difference between revisions

Revision as of 10:21, 30 May 2023

Revision as of 11:28, 30 May 2023

Line 2:

{{Redirect|Modular function|text=A distinct use of this term appears in relation to [[Haar measure#The modular function|Haar measure]]}}

In [[mathematics]], a '''modular form''' is a (complex) [[analytic function]] on the [[upper half-plane]] satisfying a certain kind of [[functional equation]] with respect to the [[Group action (mathematics)|group action]] of the [[modular group]], and also satisfying a growth condition. The theory of modular forms therefore belongs to [[complex analysis]] but the main importance of the theory has traditionally been in its connections with [[number theory]]<ref>{{Cite web |title= |url=https://mast.queensu.ca/~kani/lectures/m817w05/modforms.pdf}}</ref>. Modular forms appear in other areas, such as [[algebraic topology]], [[sphere packing]], and [[string theory]].

A '''modular function''' is a function that is invariant with respect to the modular group, but without the condition that {{math|''f''&thinsp;(''z'')}} be [[Holomorphic function|holomorphic]] in the upper half-plane (among other requirements). Instead, modular functions are [[Meromorphic function|meromorphic]] (that is, they are holomorphic on the complement of a set of isolated points, which are poles of the function).