Modular lambda function - Revision history

Bumpf at 15:53, 9 February 2025

2025-02-09T15:53:23Z

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	{{short description\|Symmetric holomorphic function}}		{{short description\|Symmetric holomorphic function}}
	[[File:Modular lambda function in range -3 to 3.png\|thumb\|Modular lambda function in the complex plane.]]		[[File:Modular lambda function in range -3 to 3.png\|thumb\|Modular lambda function in the complex plane.]]
	In [[mathematics]], the '''modular lambda''' function λ(τ)<ref group="note><math>\lambda(\tau)</math> is not a [[Modular form#Modular functions\|modular function]] (per the Wikipedia definition), but every modular function is a [[rational function]] in <math>\lambda(\tau)</math>. Some authors use a non-equivalent definition of "modular functions".</ref> is a highly symmetric [[~~Holomorphic~~ function]] on the complex [[upper half-plane]]. It is invariant under the fractional linear action of the [[congruence subgroup\|congruence group]] Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the [[modular curve]] ''X''(2). Over any point τ, its value can be described as a [[cross ratio]] of the branch points of a ramified double cover of the projective line by the [[elliptic curve]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is defined as the quotient by the [−1] involution.		In [[mathematics]], the '''modular lambda''' function λ(τ)<ref group="note><math>\lambda(\tau)</math> is not a [[Modular form#Modular functions\|modular function]] (per the Wikipedia definition), but every modular function is a [[rational function]] in <math>\lambda(\tau)</math>. Some authors use a non-equivalent definition of "modular functions".</ref> is a highly symmetric [[holomorphic function]] on the complex [[upper half-plane]]. It is invariant under the fractional linear action of the [[congruence subgroup\|congruence group]] Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the [[modular curve]] ''X''(2). Over any point τ, its value can be described as a [[cross ratio]] of the branch points of a ramified double cover of the projective line by the [[elliptic curve]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is defined as the quotient by the [−1] involution.

	The q-expansion, where <math>q = e^{\pi i \tau}</math> is the [[Nome (mathematics)\|nome]], is given by:		The q-expansion, where <math>q = e^{\pi i \tau}</math> is the [[Nome (mathematics)\|nome]], is given by:

Jengod: Short description

2024-02-03T03:21:23Z

Short description

← Previous revision		Revision as of 03:21, 3 February 2024
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			{{short description\|Symmetric holomorphic function}}
	[[File:Modular lambda function in range -3 to 3.png\|thumb\|Modular lambda function in the complex plane.]]		[[File:Modular lambda function in range -3 to 3.png\|thumb\|Modular lambda function in the complex plane.]]
	In [[mathematics]], the '''modular lambda''' function λ(τ)<ref group="note><math>\lambda(\tau)</math> is not a [[Modular form#Modular functions\|modular function]] (per the Wikipedia definition), but every modular function is a [[rational function]] in <math>\lambda(\tau)</math>. Some authors use a non-equivalent definition of "modular functions".</ref> is a highly symmetric [[Holomorphic function]] on the complex [[upper half-plane]]. It is invariant under the fractional linear action of the [[congruence subgroup\|congruence group]] Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the [[modular curve]] ''X''(2). Over any point τ, its value can be described as a [[cross ratio]] of the branch points of a ramified double cover of the projective line by the [[elliptic curve]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is defined as the quotient by the [−1] involution.		In [[mathematics]], the '''modular lambda''' function λ(τ)<ref group="note><math>\lambda(\tau)</math> is not a [[Modular form#Modular functions\|modular function]] (per the Wikipedia definition), but every modular function is a [[rational function]] in <math>\lambda(\tau)</math>. Some authors use a non-equivalent definition of "modular functions".</ref> is a highly symmetric [[Holomorphic function]] on the complex [[upper half-plane]]. It is invariant under the fractional linear action of the [[congruence subgroup\|congruence group]] Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the [[modular curve]] ''X''(2). Over any point τ, its value can be described as a [[cross ratio]] of the branch points of a ramified double cover of the projective line by the [[elliptic curve]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is defined as the quotient by the [−1] involution.

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2024-01-05T14:48:07Z

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	:<math>\lambda^*(x \in \mathbb{Q}^+) \in \mathbb{A}^+.</math>		:<math>\lambda^*(x \in \mathbb{Q}^+) \in \mathbb{A}^+.</math>

	<math>K(\lambda^(x))</math> and <math>E(\lambda^(x))</math> (the [[Elliptic integral#Complete elliptic integral of the second kind\|complete elliptic integral of the second kind]]) can be expressed in closed form in terms of the [[gamma function]] for any <math>x\in\mathbb{Q}^+</math>, as Selberg and Chowla proved in 1949.<ref>{{Cite journal\|title=On Epstein's Zeta Function (I).\|last1=Chowla\|first1=S.\|last2=Selberg\|first2=A.\|journal=Proceedings of the National Academy of Sciences \|date=1949 \|volume=35 \|issue=7 \|page=373\|doi=10.1073/PNAS.35.7.371 \|s2cid=45071481 \|doi-access=free}}</ref><ref>{{Cite web\|url=https://eudml.org/doc/150803\|title=On Epstein's Zeta-Function\|last1=Chowla\|first1=S.\|last2=Selberg\|first2=A.\|website=EuDML\|pages=86–110}}</ref>		<math>K(\lambda^(x))</math> and <math>E(\lambda^(x))</math> (the [[Elliptic integral#Complete elliptic integral of the second kind\|complete elliptic integral of the second kind]]) can be expressed in closed form in terms of the [[gamma function]] for any <math>x\in\mathbb{Q}^+</math>, as Selberg and Chowla proved in 1949.<ref>{{Cite journal\|title=On Epstein's Zeta Function (I).\|last1=Chowla\|first1=S.\|last2=Selberg\|first2=A.\|journal=Proceedings of the National Academy of Sciences \|date=1949 \|volume=35 \|issue=7 \|page=373\|doi=10.1073/PNAS.35.7.371 \|s2cid=45071481 \|doi-access=free\|pmc=1063041}}</ref><ref>{{Cite web\|url=https://eudml.org/doc/150803\|title=On Epstein's Zeta-Function\|last1=Chowla\|first1=S.\|last2=Selberg\|first2=A.\|website=EuDML\|pages=86–110}}</ref>

	The following expression is valid for all <math>n \in \mathbb{N}</math>:		The following expression is valid for all <math>n \in \mathbb{N}</math>:

OAbot: Open access bot: doi updated in citation with #oabot.

2023-11-08T14:18:57Z

Open access bot: doi updated in citation with #oabot.

← Previous revision		Revision as of 14:18, 8 November 2023
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	:<math>\lambda^*(x \in \mathbb{Q}^+) \in \mathbb{A}^+.</math>		:<math>\lambda^*(x \in \mathbb{Q}^+) \in \mathbb{A}^+.</math>

	<math>K(\lambda^(x))</math> and <math>E(\lambda^(x))</math> (the [[Elliptic integral#Complete elliptic integral of the second kind\|complete elliptic integral of the second kind]]) can be expressed in closed form in terms of the [[gamma function]] for any <math>x\in\mathbb{Q}^+</math>, as Selberg and Chowla proved in 1949.<ref>{{Cite journal\|title=On Epstein's Zeta Function (I).\|last1=Chowla\|first1=S.\|last2=Selberg\|first2=A.\|journal=Proceedings of the National Academy of Sciences \|date=1949 \|volume=35 \|issue=7 \|page=373\|doi=10.1073/PNAS.35.7.371 \|s2cid=45071481 }}</ref><ref>{{Cite web\|url=https://eudml.org/doc/150803\|title=On Epstein's Zeta-Function\|last1=Chowla\|first1=S.\|last2=Selberg\|first2=A.\|website=EuDML\|pages=86–110}}</ref>		<math>K(\lambda^(x))</math> and <math>E(\lambda^(x))</math> (the [[Elliptic integral#Complete elliptic integral of the second kind\|complete elliptic integral of the second kind]]) can be expressed in closed form in terms of the [[gamma function]] for any <math>x\in\mathbb{Q}^+</math>, as Selberg and Chowla proved in 1949.<ref>{{Cite journal\|title=On Epstein's Zeta Function (I).\|last1=Chowla\|first1=S.\|last2=Selberg\|first2=A.\|journal=Proceedings of the National Academy of Sciences \|date=1949 \|volume=35 \|issue=7 \|page=373\|doi=10.1073/PNAS.35.7.371 \|s2cid=45071481 \|doi-access=free}}</ref><ref>{{Cite web\|url=https://eudml.org/doc/150803\|title=On Epstein's Zeta-Function\|last1=Chowla\|first1=S.\|last2=Selberg\|first2=A.\|website=EuDML\|pages=86–110}}</ref>

	The following expression is valid for all <math>n \in \mathbb{N}</math>:		The following expression is valid for all <math>n \in \mathbb{N}</math>:

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2023-10-25T10:05:42Z

Alter: template type. Add: pages, s2cid, doi, issue, volume, date, journal. Removed proxy/dead URL that duplicated identifier. Removed parameters. Some additions/deletions were parameter name changes. | Use this bot. Report bugs. | #UCB_CommandLine

← Previous revision		Revision as of 10:05, 25 October 2023
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	:<math>\lambda^*(x \in \mathbb{Q}^+) \in \mathbb{A}^+.</math>		:<math>\lambda^*(x \in \mathbb{Q}^+) \in \mathbb{A}^+.</math>

	<math>K(\lambda^(x))</math> and <math>E(\lambda^(x))</math> (the [[Elliptic integral#Complete elliptic integral of the second kind\|complete elliptic integral of the second kind]]) can be expressed in closed form in terms of the [[gamma function]] for any <math>x\in\mathbb{Q}^+</math>, as Selberg and Chowla proved in 1949.<ref>{{Cite ~~web\|url=https://www.semanticscholar.org/paper/On-Epstein's-Zeta-Function-(I).-Chowla-Selberg/87dc02200853b431bfa900e297cd6c2f80a5a4b1~~\|title=On Epstein's Zeta Function (I).\|last1=Chowla\|first1=S.\|last2=Selberg\|first2=A.\|~~website~~=~~Semantic~~ ~~Scholar~~\|page=373}}</ref><ref>{{Cite web\|url=https://eudml.org/doc/150803\|title=On Epstein's Zeta-Function\|last1=Chowla\|first1=S.\|last2=Selberg\|first2=A.\|website=EuDML\|~~page~~=86–110}}</ref>		<math>K(\lambda^(x))</math> and <math>E(\lambda^(x))</math> (the [[Elliptic integral#Complete elliptic integral of the second kind\|complete elliptic integral of the second kind]]) can be expressed in closed form in terms of the [[gamma function]] for any <math>x\in\mathbb{Q}^+</math>, as Selberg and Chowla proved in 1949.<ref>{{Cite journal\|title=On Epstein's Zeta Function (I).\|last1=Chowla\|first1=S.\|last2=Selberg\|first2=A.\|journal=Proceedings of the National Academy of Sciences \|date=1949 \|volume=35 \|issue=7 \|page=373\|doi=10.1073/PNAS.35.7.371 \|s2cid=45071481 }}</ref><ref>{{Cite web\|url=https://eudml.org/doc/150803\|title=On Epstein's Zeta-Function\|last1=Chowla\|first1=S.\|last2=Selberg\|first2=A.\|website=EuDML\|pages=86–110}}</ref>

	The following expression is valid for all <math>n \in \mathbb{N}</math>:		The following expression is valid for all <math>n \in \mathbb{N}</math>:

2601:483:800:2EC0:315F:6339:675B:5E4A: Added links

2023-06-29T01:36:32Z

Added links

← Previous revision		Revision as of 01:36, 29 June 2023
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	[[File:Modular lambda function in range -3 to 3.png\|thumb\|Modular lambda function in the complex plane.]]		[[File:Modular lambda function in range -3 to 3.png\|thumb\|Modular lambda function in the complex plane.]]
	In [[mathematics]], the '''modular lambda''' function λ(τ)<ref group="note><math>\lambda(\tau)</math> is not a [[Modular form#Modular functions\|modular function]] (per the Wikipedia definition), but every modular function is a [[rational function]] in <math>\lambda(\tau)</math>. Some authors use a non-equivalent definition of "modular functions".</ref> is a highly symmetric ~~holomorphic~~ function on the complex [[upper half-plane]]. It is invariant under the fractional linear action of the [[congruence subgroup\|congruence group]] Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the [[modular curve]] ''X''(2). Over any point τ, its value can be described as a [[cross ratio]] of the branch points of a ramified double cover of the projective line by the [[elliptic curve]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is defined as the quotient by the [−1] involution.		In [[mathematics]], the '''modular lambda''' function λ(τ)<ref group="note><math>\lambda(\tau)</math> is not a [[Modular form#Modular functions\|modular function]] (per the Wikipedia definition), but every modular function is a [[rational function]] in <math>\lambda(\tau)</math>. Some authors use a non-equivalent definition of "modular functions".</ref> is a highly symmetric [[Holomorphic function]] on the complex [[upper half-plane]]. It is invariant under the fractional linear action of the [[congruence subgroup\|congruence group]] Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the [[modular curve]] ''X''(2). Over any point τ, its value can be described as a [[cross ratio]] of the branch points of a ramified double cover of the projective line by the [[elliptic curve]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is defined as the quotient by the [−1] involution.

	The q-expansion, where <math>q = e^{\pi i \tau}</math> is the [[Nome (mathematics)\|nome]], is given by:		The q-expansion, where <math>q = e^{\pi i \tau}</math> is the [[Nome (mathematics)\|nome]], is given by:

A1E6: /* Relations to other functions */

2023-01-01T20:39:47Z

Relations to other functions

← Previous revision		Revision as of 20:39, 1 January 2023
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	which is the ''j''-invariant of the elliptic curve of [[Legendre form]] <math>y^2=x(x-1)(x-\lambda)</math>		which is the ''j''-invariant of the elliptic curve of [[Legendre form]] <math>y^2=x(x-1)(x-\lambda)</math>

			Given <math>m\in\mathbb{C}\setminus\{0,1\}</math>, let
			:<math>\tau=i\frac{K\{1-m\}}{K\{m\}}</math>
			where <math>K</math> is the [[Elliptic integral#Complete elliptic integral of the first kind\|complete elliptic integral of the first kind]] with parameter <math>m=k^2</math>.
			Then
			:<math>\lambda (\tau)=m.</math>

	==Modular equations==		==Modular equations==

2A02:842B:80F5:1A01:45CE:AF02:1C5F:B21A: /* Properties of lambda-star */ better spacing

2022-12-08T21:31:26Z

Properties of lambda-star: better spacing

← Previous revision		Revision as of 21:31, 8 December 2022
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	:<math>\lambda^(x) - \lambda^(9x) = 2[\lambda^(x)\lambda^(9x)]^{1/4} - 2[\lambda^(x)\lambda^(9x)]^{3/4}</math>		:<math>\lambda^(x) - \lambda^(9x) = 2[\lambda^(x)\lambda^(9x)]^{1/4} - 2[\lambda^(x)\lambda^(9x)]^{3/4}</math>

			<math display=block>\begin{align}
		⚫	& a^{6}-f^{6} = 2af +2a^5f^5\, &\left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) &\left(f = \left[\frac{2\lambda^(25x)}{1-\lambda^(25x)^2}\right]^{1/12}\right) \\
		⚫	&a^{8}+b^{8}-7a^4b^4 = 2\sqrt{2}ab+2\sqrt{2}a^7b^7\, &\left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) &\left(b = \left[\frac{2\lambda^(49x)}{1-\lambda^(49x)^2}\right]^{1/12}\right) \\

	~~:<math>~~a^{6}-f^{6} = ~~2af~~ +2a^5f^5\, \left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) \left(f = \left[\frac{2\lambda^(~~25x~~)}{1-\lambda^(~~25x~~)^2}\right]^{1/12}\right) ~~</math>~~		& a^{12}-c^{12} = 2\sqrt{2}(ac+a^3c^3)(1+3a^2c^2+a^4c^4)(2+3a^2c^2+2a^4c^4)\, &\left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) &\left(c = \left[\frac{2\lambda^(121x)}{1-\lambda^(121x)^2}\right]^{1/12}\right) \\

	~~:<math>~~a^~~{8}~~+b^~~{8}~~-7a^4b^4 = 2~~\sqrt{~~2~~}ab~~+2~~\sqrt{~~2}a^~~7b^7~~\, \left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) \left(b = \left[\frac{2\lambda^(~~49x~~)}{1-\lambda^(~~49x~~)^2}\right]^{1/12}\right) ~~</math>~~		& (a^2-d^2)(a^4+d^4-7a^2d^2)[(a^2-d^2)^4-a^2d^2(a^2+d^2)^2] = 8ad+8a^{13}d^{13}\, &\left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) &\left(d = \left[\frac{2\lambda^(169x)}{1-\lambda^(169x)^2}\right]^{1/12}\right)
			\end{align}

			</math>
⚫	~~:<math>~~a^{12}-c^{12} = ~~2\sqrt{2}(ac+a^3c^3)(1+3a^2c^2+a^4c^4)(2+3a^2c^2~~+2a^4c^4)\, \left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) \left(c = \left[\frac{2\lambda^(~~121x~~)}{1-\lambda^(~~121x~~)^2}\right]^{1/12}\right) ~~</math>~~

⚫	~~:<math>(~~a^~~2-d^2)(a^4~~+d^4-7a^~~2d^2)[(a^2-d^2)~~^4~~-a^2d^~~2~~(a^~~2+d^2)^2~~] = 8ad+8a^{13~~}d^~~{13}~~\, \left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) \left(d = \left[\frac{2\lambda^(~~169x~~)}{1-\lambda^(~~169x~~)^2}\right]^{1/12}\right) ~~</math>~~

	{{Collapse top\|title=Special values}}		{{Collapse top\|title=Special values}}

2A02:842B:80F5:1A01:45CE:AF02:1C5F:B21A: /* Properties of lambda-star */

2022-12-08T21:20:32Z

Properties of lambda-star

← Previous revision		Revision as of 21:20, 8 December 2022
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	:<math>\lambda^(x) - \lambda^(9x) = 2[\lambda^(x)\lambda^(9x)]^{1/4} - 2[\lambda^(x)\lambda^(9x)]^{3/4}</math>		:<math>\lambda^(x) - \lambda^(9x) = 2[\lambda^(x)\lambda^(9x)]^{1/4} - 2[\lambda^(x)\lambda^(9x)]^{3/4}</math>


	:<math>\left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/2} - \left[\frac{2\lambda^(25x)}{1-\lambda^(25x)^2}\right]^{1/2} = 2\left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\left[\frac{2\lambda^(25x)}{1-\lambda^(25x)^2}\right]^{1/12} + 2\left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{5/12}\left[\frac{2\lambda^(25x)}{1-\lambda^(25x)^2}\right]^{5/12} </math>
			:<math>a^{6}-f^{6} = 2af +2a^5f^5\, \left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) \left(f = \left[\frac{2\lambda^(25x)}{1-\lambda^(25x)^2}\right]^{1/12}\right) </math>

	:<math>a^{8}+b^{8}-7a^4b^4 = 2\sqrt{2}ab+2\sqrt{2}a^7b^7\, \left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) \left(b = \left[\frac{2\lambda^(49x)}{1-\lambda^(49x)^2}\right]^{1/12}\right) </math>		:<math>a^{8}+b^{8}-7a^4b^4 = 2\sqrt{2}ab+2\sqrt{2}a^7b^7\, \left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) \left(b = \left[\frac{2\lambda^(49x)}{1-\lambda^(49x)^2}\right]^{1/12}\right) </math>

A1E6: /* Modular equations */

2022-07-06T20:04:57Z

Modular equations

← Previous revision		Revision as of 20:04, 6 July 2022
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	:<math>\lambda (ni)=\prod_{k=1}^{n/2} \operatorname{sl}^8\frac{(2k-1)\varpi}{2n}\quad (n\,\text{even})</math>		:<math>\lambda (ni)=\prod_{k=1}^{n/2} \operatorname{sl}^8\frac{(2k-1)\varpi}{2n}\quad (n\,\text{even})</math>
	:<math>\lambda (ni)=\frac{1}{2^n}\prod_{k=1}^{n-1} \left(1-\operatorname{sl}^2\frac{k\varpi}{n}\right)^2\quad (n\,\text{odd})</math>		:<math>\lambda (ni)=\frac{1}{2^n}\prod_{k=1}^{n-1} \left(1-\operatorname{sl}^2\frac{k\varpi}{n}\right)^2\quad (n\,\text{odd})</math>
	where <math>\operatorname{sl}</math> is the [[Lemniscate elliptic functions\|lemniscate sine]] and <math>\varpi</math> is the [[~~Lemniscate elliptic functions#Lemniscate constant\|~~lemniscate constant]].		where <math>\operatorname{sl}</math> is the [[Lemniscate elliptic functions\|lemniscate sine]] and <math>\varpi</math> is the [[lemniscate constant]].

	==Lambda-star==		==Lambda-star==

← Previous revision		Revision as of 15:53, 9 February 2025
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	{{short description\|Symmetric holomorphic function}}		{{short description\|Symmetric holomorphic function}}
	[[File:Modular lambda function in range -3 to 3.png\|thumb\|Modular lambda function in the complex plane.]]		[[File:Modular lambda function in range -3 to 3.png\|thumb\|Modular lambda function in the complex plane.]]
	In [[mathematics]], the '''modular lambda''' function λ(τ)<ref group="note><math>\lambda(\tau)</math> is not a [[Modular form#Modular functions\|modular function]] (per the Wikipedia definition), but every modular function is a [[rational function]] in <math>\lambda(\tau)</math>. Some authors use a non-equivalent definition of "modular functions".</ref> is a highly symmetric [[~~Holomorphic~~ function]] on the complex [[upper half-plane]]. It is invariant under the fractional linear action of the [[congruence subgroup\|congruence group]] Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the [[modular curve]] ''X''(2). Over any point τ, its value can be described as a [[cross ratio]] of the branch points of a ramified double cover of the projective line by the [[elliptic curve]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is defined as the quotient by the [−1] involution.		In [[mathematics]], the '''modular lambda''' function λ(τ)<ref group="note><math>\lambda(\tau)</math> is not a [[Modular form#Modular functions\|modular function]] (per the Wikipedia definition), but every modular function is a [[rational function]] in <math>\lambda(\tau)</math>. Some authors use a non-equivalent definition of "modular functions".</ref> is a highly symmetric [[holomorphic function]] on the complex [[upper half-plane]]. It is invariant under the fractional linear action of the [[congruence subgroup\|congruence group]] Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the [[modular curve]] ''X''(2). Over any point τ, its value can be described as a [[cross ratio]] of the branch points of a ramified double cover of the projective line by the [[elliptic curve]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is defined as the quotient by the [−1] involution.

	The q-expansion, where <math>q = e^{\pi i \tau}</math> is the [[Nome (mathematics)\|nome]], is given by:		The q-expansion, where <math>q = e^{\pi i \tau}</math> is the [[Nome (mathematics)\|nome]], is given by:

← Previous revision		Revision as of 03:21, 3 February 2024
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			{{short description\|Symmetric holomorphic function}}
	[[File:Modular lambda function in range -3 to 3.png\|thumb\|Modular lambda function in the complex plane.]]		[[File:Modular lambda function in range -3 to 3.png\|thumb\|Modular lambda function in the complex plane.]]
	In [[mathematics]], the '''modular lambda''' function λ(τ)<ref group="note><math>\lambda(\tau)</math> is not a [[Modular form#Modular functions\|modular function]] (per the Wikipedia definition), but every modular function is a [[rational function]] in <math>\lambda(\tau)</math>. Some authors use a non-equivalent definition of "modular functions".</ref> is a highly symmetric [[Holomorphic function]] on the complex [[upper half-plane]]. It is invariant under the fractional linear action of the [[congruence subgroup\|congruence group]] Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the [[modular curve]] ''X''(2). Over any point τ, its value can be described as a [[cross ratio]] of the branch points of a ramified double cover of the projective line by the [[elliptic curve]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is defined as the quotient by the [−1] involution.		In [[mathematics]], the '''modular lambda''' function λ(τ)<ref group="note><math>\lambda(\tau)</math> is not a [[Modular form#Modular functions\|modular function]] (per the Wikipedia definition), but every modular function is a [[rational function]] in <math>\lambda(\tau)</math>. Some authors use a non-equivalent definition of "modular functions".</ref> is a highly symmetric [[Holomorphic function]] on the complex [[upper half-plane]]. It is invariant under the fractional linear action of the [[congruence subgroup\|congruence group]] Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the [[modular curve]] ''X''(2). Over any point τ, its value can be described as a [[cross ratio]] of the branch points of a ramified double cover of the projective line by the [[elliptic curve]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is defined as the quotient by the [−1] involution.

← Previous revision		Revision as of 01:36, 29 June 2023
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	[[File:Modular lambda function in range -3 to 3.png\|thumb\|Modular lambda function in the complex plane.]]		[[File:Modular lambda function in range -3 to 3.png\|thumb\|Modular lambda function in the complex plane.]]
	In [[mathematics]], the '''modular lambda''' function λ(τ)<ref group="note><math>\lambda(\tau)</math> is not a [[Modular form#Modular functions\|modular function]] (per the Wikipedia definition), but every modular function is a [[rational function]] in <math>\lambda(\tau)</math>. Some authors use a non-equivalent definition of "modular functions".</ref> is a highly symmetric ~~holomorphic~~ function on the complex [[upper half-plane]]. It is invariant under the fractional linear action of the [[congruence subgroup\|congruence group]] Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the [[modular curve]] ''X''(2). Over any point τ, its value can be described as a [[cross ratio]] of the branch points of a ramified double cover of the projective line by the [[elliptic curve]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is defined as the quotient by the [−1] involution.		In [[mathematics]], the '''modular lambda''' function λ(τ)<ref group="note><math>\lambda(\tau)</math> is not a [[Modular form#Modular functions\|modular function]] (per the Wikipedia definition), but every modular function is a [[rational function]] in <math>\lambda(\tau)</math>. Some authors use a non-equivalent definition of "modular functions".</ref> is a highly symmetric [[Holomorphic function]] on the complex [[upper half-plane]]. It is invariant under the fractional linear action of the [[congruence subgroup\|congruence group]] Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the [[modular curve]] ''X''(2). Over any point τ, its value can be described as a [[cross ratio]] of the branch points of a ramified double cover of the projective line by the [[elliptic curve]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is defined as the quotient by the [−1] involution.

	The q-expansion, where <math>q = e^{\pi i \tau}</math> is the [[Nome (mathematics)\|nome]], is given by:		The q-expansion, where <math>q = e^{\pi i \tau}</math> is the [[Nome (mathematics)\|nome]], is given by:

← Previous revision		Revision as of 21:31, 8 December 2022
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	:<math>\lambda^(x) - \lambda^(9x) = 2[\lambda^(x)\lambda^(9x)]^{1/4} - 2[\lambda^(x)\lambda^(9x)]^{3/4}</math>		:<math>\lambda^(x) - \lambda^(9x) = 2[\lambda^(x)\lambda^(9x)]^{1/4} - 2[\lambda^(x)\lambda^(9x)]^{3/4}</math>

			<math display=block>\begin{align}
		⚫	& a^{6}-f^{6} = 2af +2a^5f^5\, &\left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) &\left(f = \left[\frac{2\lambda^(25x)}{1-\lambda^(25x)^2}\right]^{1/12}\right) \\
		⚫	&a^{8}+b^{8}-7a^4b^4 = 2\sqrt{2}ab+2\sqrt{2}a^7b^7\, &\left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) &\left(b = \left[\frac{2\lambda^(49x)}{1-\lambda^(49x)^2}\right]^{1/12}\right) \\

	~~:<math>~~a^{6}-f^{6} = ~~2af~~ +2a^5f^5\, \left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) \left(f = \left[\frac{2\lambda^(~~25x~~)}{1-\lambda^(~~25x~~)^2}\right]^{1/12}\right) ~~</math>~~		& a^{12}-c^{12} = 2\sqrt{2}(ac+a^3c^3)(1+3a^2c^2+a^4c^4)(2+3a^2c^2+2a^4c^4)\, &\left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) &\left(c = \left[\frac{2\lambda^(121x)}{1-\lambda^(121x)^2}\right]^{1/12}\right) \\

	~~:<math>~~a^~~{8}~~+b^~~{8}~~-7a^4b^4 = 2~~\sqrt{~~2~~}ab~~+2~~\sqrt{~~2}a^~~7b^7~~\, \left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) \left(b = \left[\frac{2\lambda^(~~49x~~)}{1-\lambda^(~~49x~~)^2}\right]^{1/12}\right) ~~</math>~~		& (a^2-d^2)(a^4+d^4-7a^2d^2)[(a^2-d^2)^4-a^2d^2(a^2+d^2)^2] = 8ad+8a^{13}d^{13}\, &\left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) &\left(d = \left[\frac{2\lambda^(169x)}{1-\lambda^(169x)^2}\right]^{1/12}\right)
			\end{align}

			</math>
⚫	~~:<math>~~a^{12}-c^{12} = ~~2\sqrt{2}(ac+a^3c^3)(1+3a^2c^2+a^4c^4)(2+3a^2c^2~~+2a^4c^4)\, \left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) \left(c = \left[\frac{2\lambda^(~~121x~~)}{1-\lambda^(~~121x~~)^2}\right]^{1/12}\right) ~~</math>~~

⚫	~~:<math>(~~a^~~2-d^2)(a^4~~+d^4-7a^~~2d^2)[(a^2-d^2)~~^4~~-a^2d^~~2~~(a^~~2+d^2)^2~~] = 8ad+8a^{13~~}d^~~{13}~~\, \left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) \left(d = \left[\frac{2\lambda^(~~169x~~)}{1-\lambda^(~~169x~~)^2}\right]^{1/12}\right) ~~</math>~~

	{{Collapse top\|title=Special values}}		{{Collapse top\|title=Special values}}

← Previous revision		Revision as of 21:20, 8 December 2022
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	:<math>\lambda^(x) - \lambda^(9x) = 2[\lambda^(x)\lambda^(9x)]^{1/4} - 2[\lambda^(x)\lambda^(9x)]^{3/4}</math>		:<math>\lambda^(x) - \lambda^(9x) = 2[\lambda^(x)\lambda^(9x)]^{1/4} - 2[\lambda^(x)\lambda^(9x)]^{3/4}</math>


	:<math>\left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/2} - \left[\frac{2\lambda^(25x)}{1-\lambda^(25x)^2}\right]^{1/2} = 2\left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\left[\frac{2\lambda^(25x)}{1-\lambda^(25x)^2}\right]^{1/12} + 2\left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{5/12}\left[\frac{2\lambda^(25x)}{1-\lambda^(25x)^2}\right]^{5/12} </math>
			:<math>a^{6}-f^{6} = 2af +2a^5f^5\, \left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) \left(f = \left[\frac{2\lambda^(25x)}{1-\lambda^(25x)^2}\right]^{1/12}\right) </math>

	:<math>a^{8}+b^{8}-7a^4b^4 = 2\sqrt{2}ab+2\sqrt{2}a^7b^7\, \left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) \left(b = \left[\frac{2\lambda^(49x)}{1-\lambda^(49x)^2}\right]^{1/12}\right) </math>		:<math>a^{8}+b^{8}-7a^4b^4 = 2\sqrt{2}ab+2\sqrt{2}a^7b^7\, \left(a = \left[\frac{2\lambda^(x)}{1-\lambda^(x)^2}\right]^{1/12}\right) \left(b = \left[\frac{2\lambda^(49x)}{1-\lambda^(49x)^2}\right]^{1/12}\right) </math>