Modular function: Difference between revisions
Content deleted Content added
Laurent -> Fourier |
Merge, as suggested. |
||
| Line 1: | Line 1: | ||
#REDIRECT [[Modular form]] |
|||
In [[mathematics]], '''modular functions''' are certain kinds of [[function (mathematics)|mathematical function]]s mapping [[complex number]]s to complex numbers. There are a number of other uses of the term "modular function" as well; see below for details. |
|||
Formally, a function ''f'' is called '''modular''' or a '''modular function''' [[iff]] it satisfies the following properties: |
|||
# ''f'' is [[meromorphic function|meromorphic]] in the open [[upper half-plane]] ''H''. |
|||
# For every [[matrix (mathematics)|matrix]] ''M'' in the [[modular group Gamma|modular group Γ]], ''f''(''M''τ) = ''f''(τ). |
|||
# The [[Fourier series]] of ''f'' has the form |
|||
::<math>f(\tau) = \sum_{n=-m}^\infty a(n) e^{2i\pi n\tau}.</math> |
|||
It is bounded below; it is a [[Laurent polynomial]] in <math>e^{2i\pi \tau}</math>, so it is meromorphic at the cusp. |
|||
It can be shown that every modular function can be expressed as a [[rational function]] of [[Klein's absolute invariant]] ''j''(τ), and that every rational function of ''j''(τ) is a modular function; furthermore, all [[analytic function|analytic]] modular functions are [[modular form]]s, although the converse does not hold. If a modular function ''f'' is not identically 0, then it can be shown that the number of zeroes of ''f'' is equal to the number of [[pole (complex analysis)|pole]]s of ''f'' in the [[closure (mathematics)|closure]] of the [[fundamental region]] ''R''<sub>Γ</sub>. |
|||
== Other uses == |
|||
There are a number of other usages of the term '''''modular function''''', apart from this classical one; for example, in the theory of [[Haar measure]]s, it is a function Δ(''g'') determined by the conjugation action. |
|||
== References == |
|||
* Tom M. Apostol, ''Modular functions and Dirichlet Series in Number Theory'' (1990), Springer-Verlag, New York. ISBN 0-387-97127-0 |
|||
* Robert A. Rankin, ''Modular forms and functions'', (1977) Cambridge University Press, Cambridge. ISBN 0-521-21212-X |
|||
[[Category:Moduli theory]] |
|||
[[Category:Special functions]] |
|||
[[ru:Модулярная функция]] |
|||
{{math-stub}} |
|||
Revision as of 19:06, 20 October 2007
Redirect to: