Modular function - Revision history

1234qwer1234qwer4: Added {{R from merge}} tag to redirect

2024-08-02T00:02:48Z

Added {{R from merge}} tag to redirect

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Nbarth: Merge, as suggested.

2007-10-20T19:06:06Z

Merge, as suggested.

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	~~{{Mergeto\|~~Modular form~~\|date=March 2007}}~~		#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}}

Nbarth: Laurent -> Fourier

2007-10-20T18:35:00Z

Laurent -> Fourier

← Previous revision		Revision as of 18:35, 20 October 2007
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	# ''f'' is [[meromorphic function\|meromorphic]] in the open [[upper half-plane]] ''H''.		# ''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).		# For every [[matrix (mathematics)\|matrix]] ''M'' in the [[modular group Gamma\|modular group Γ]], ''f''(''M''τ) = ''f''(τ).
	# The [[~~Laurent~~ series]] of ''f'' has the form		# The [[Fourier series]] of ''f'' has the form

	::<math>f(\tau) = \sum_{n=-m}^\infty a(n) e^{2i\pi n\tau}.</math>		::<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>.		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>.

SmackBot: Date/fix maintenance tags

2007-03-29T13:48:44Z

Date/fix maintenance tags

← Previous revision		Revision as of 13:48, 29 March 2007
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	{{~~mergeto~~\|Modular form}}		{{Mergeto\|Modular form\|date=March 2007}}
	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.		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.

Arcfrk: added merge tag

2007-03-29T05:49:51Z

added merge tag

← Previous revision		Revision as of 05:49, 29 March 2007
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			{{mergeto\|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.		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.

SmackBot: ISBN formatting/gen fixes using AWB

2006-11-22T21:52:40Z

ISBN formatting/gen fixes using AWB

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	* Tom M. Apostol, ''Modular functions and Dirichlet Series in Number Theory'' (1990), Springer-Verlag, New York. ISBN 0-387-97127-0		* 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		* Robert A. Rankin, ''Modular forms and functions'', (1977) Cambridge University Press, Cambridge. ISBN 0-521-21212-X

⚫	{{math-stub}}

	[[Category:Moduli theory]]		[[Category:Moduli theory]]
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	[[ru:Модулярная функция]]		[[ru:Модулярная функция]]


		⚫	{{math-stub}}

82.240.108.85 at 14:25, 30 August 2006

2006-08-30T14:25:12Z

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	Formally, a function ''f'' is called '''modular''' or a '''modular function''' [[iff]] it satisfies the following properties:		Formally, a function ''f'' is called '''modular''' or a '''modular function''' [[iff]] it satisfies the following properties:

	# ''f'' is [[meromorphic function\|meromorphic]] in the [[upper half-plane]] ''H''.		# ''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).		# For every [[matrix (mathematics)\|matrix]] ''M'' in the [[modular group Gamma\|modular group Γ]], ''f''(''M''τ) = ''f''(τ).
	# The [[Laurent series]] of ''f'' has the form		# The [[Laurent series]] of ''f'' has the form

YurikBot: robot Adding: ru:Модулярная функция

2006-04-08T04:56:59Z

robot Adding: ru:Модулярная функция

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	[[Category:Moduli theory]]		[[Category:Moduli theory]]
	[[Category:Special functions]]		[[Category:Special functions]]

			[[ru:Модулярная функция]]

Michael Hardy at 20:36, 9 October 2005

2005-10-09T20:36:31Z

← Previous revision		Revision as of 20:36, 9 October 2005
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	In [[mathematics]], '''modular functions''' are certain kinds of [[function (mathematics)\|mathematical function]]s mapping [[complex ~~numbers~~]] to complex numbers. There are a number of other uses of the term "modular function" as well; see below for details.		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:		Formally, a function ''f'' is called '''modular''' or a '''modular function''' [[iff]] it satisfies the following properties:

	# ''f'' is [[meromorphic function\|meromorphic]] in the [[upper half plane]] ''H''.		# ''f'' is [[meromorphic function\|meromorphic]] in the [[upper half-plane]] ''H''.
	# For every [[matrix (mathematics)\|matrix]] ''M'' in the [[modular group Gamma\|modular group Γ]], ''f''(''M''τ) = ''f''(τ).		# For every [[matrix (mathematics)\|matrix]] ''M'' in the [[modular group Gamma\|modular group Γ]], ''f''(''M''τ) = ''f''(τ).
	# The [[Laurent series]] of ''f'' has the form		# The [[Laurent series]] of ''f'' has the form

Charles Matthews at 20:15, 28 September 2005

2005-09-28T20:15:30Z

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	[[Category:~~Modular~~ ~~forms~~]]		[[Category:Moduli theory]]
			[[Category:Special functions]]