Minkowski's question mark function - Revision history

84.32.73.63: /* References */

2014-11-29T12:34:44Z

References

← Older revision		Revision as of 14:34, 29 November 2014
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	~~In [[mathematics]], the '''Jacobi triple product''' is~~ the ~~mathematical identity:~~		Hello. Let me introduce the author. Her title is Emilia Shroyer but it's not the most feminine name out there. I am a meter reader but I plan on changing it. Her family members lives in Minnesota. To perform baseball is the hobby he will by no means quit performing.<br><br>My web-site; over the counter std test ([http://sanamaq.kz/ru/blogs/clear-candida-these-tips continue reading this..])

	~~:<math>\prod_{m=1}^\infty~~
	~~\left( 1 - x^{2m}\right)~~
	~~\left( 1 + x^{2m-1} y^2\right)~~
	~~\left( 1 + x^{2m-1} y^{-2}\right)~~
	~~= \sum_{n=-\infty}^\infty x^{n^2} y^{2n},~~
	~~</math>~~
	~~for complex numbers ''x'' and ''y'', with \|''x''\| < 1 and ''y'' ≠ 0~~.

	~~It was introduced by {{harvs\|txt\|authorlink=Carl Gustav Jacob Jacobi\|last=Jacobi\|year=1829}} in his work ''[[Fundamenta Nova Theoriae Functionum Ellipticarum]]''.~~

	~~The Jacobi triple product identity~~ is ~~the [[Macdonald identity]] for the affine root system of type~~ '~~'A''<sub>1</sub>, and is the [[Weyl denominator formula]] for~~ the ~~corresponding affine [[Kac–Moody algebra]]~~.

	~~== Properties ==~~

	~~The basis of Jacobi's proof relies on Euler's [[pentagonal number theorem]], which is itself~~ a ~~specific case of the Jacobi Triple Product Identity.~~

	~~Let <math>x=q^{3/2}</math> and <math>y^2=-\sqrt{q}</math>. Then we have~~
	~~:<math>\phi(q) = \prod_{m=1}^\infty \left(1-q^m \right) =~~
	~~\sum_{n=-\infty}^\infty (-1)^n q^{(3n^2-n)/2}.\, </math>~~

	~~The Jacobi Triple Product also allows the Jacobi [[theta function]] to be written as an infinite product as follows:~~

	~~Let <math>x=e^{i\pi \tau}</math> and <math>y=e^{i\pi z}.</math>~~

	~~Then the Jacobi theta function~~

	~~:<math>~~
	~~\vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp (\pi i n^2 \tau + 2 \pi i n z)~~
	~~</math>~~

	~~can be written in the form~~

	~~:<math>\sum_{n=-\infty}^\infty y^{2n}x^{n^2}. </math>~~

	~~Using the Jacobi Triple Product Identity we can then write the theta function as the product~~

	~~:<math>\vartheta(z; \tau) = \prod_{m=1}^\infty~~
	~~\left( 1 - \exp(2m \pi i \tau)\right)~~
	~~\left( 1 + \exp((2m-1) \pi i \tau + 2 \pi i z)\right)~~
	~~\left( 1 + \exp((2m-1) \pi i \tau -2 \pi i z)\right).~~
	~~</math>~~

	~~There are many different notations used to express the Jacobi triple product. It takes~~ on ~~a concise form when expressed in terms of [[q-Pochhammer symbol]]s:~~

	~~:<math>\sum_{n=-\infty}^\infty q^{n(n+1)/2}z^n =~~
	~~(q;q)_\infty \; (-1/z;q)_\infty \; (-zq;q)_\infty.</math>~~

	~~Where <math>(a;q)_\infty</math> is the infinite ''q''-Pochhammer symbol.~~

	~~It enjoys a particularly elegant form when expressed in terms of the [[Ramanujan theta function]]. For <math>\|ab\|<1.</math>~~ it ~~can be written as~~

	~~:<math>\sum_{n=-\infty}^\infty a^{n(n+1)/2} \; b^{n(n-1)/2} = (-a; ab)_\infty \;(-b; ab)_\infty \;(ab;ab)_\infty.</math>~~

	~~==Proof==~~
	~~This proof uses a simplified model of the [[Dirac sea]] and follows the proof in Cameron (13.3) which is attributed to [[Richard Borcherds]]~~. ~~It treats the case where the power series are formal. For the analytic case, see Apostol. The Jacobi triple product identity can be expressed as~~

	~~:<math>\prod_{n>0}(1+q^{n-\frac{1}{2}}z)(1+q^{n-\frac{1}{2}}z^{-1})=\left(\sum_{l\~~in~~\mathbb{Z}}q^{l^2/2}z^l\right)\left(\prod_{n>0}(1-q^n)^{-1}\right).</math>~~

	~~A ''level'' is a [[half-integer]]. The vacuum state is the set of all negative levels. A state is a set of levels whose symmetric difference with the vacuum state is finite~~. ~~The ''energy'' of the state <math>S</math> is~~

	~~:<math>\sum\{v\colon v > 0,v\in S\} - \sum\{v\colon v < 0, v\not\in S\}</math>~~

	~~and the ''particle number'' of <math>S</math>~~ is

	~~:<math>\|\{v\colon v>0,v\in S\}\|-\|\{v\colon v<0,v\not\in S\}\|.</math>~~

	~~An unordered choice of~~ the presence of finitely many positive levels and the absence of finitely many negative levels (relative to the vacuum) corresponds to a state, so the generating function <math>\textstyle\sum_{m,l} s(m,l)q^mz^l</math> for the number <math>s(m,l)</math> of states of energy <math>m</math> with <math>l</math> particles can be expressed as

	~~:<math>\prod_{n>0}(1+q^{n-\frac{1}{2}}z)(1+q^{n-\frac{1}{2}}z^{-1}).</math>~~

	~~On the other hand, any state with <math>l</math> particles can be obtained from the lowest energy <math>l-</math>particle state, <math>\{v\colon v<l\}</math>,~~ by rearranging particles: take a partition <math>\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_j</math> of <math>m'</math> and move the top particle up by <math>\lambda_1</math> levels, the next highest particle up by <math>\lambda_2</math> levels, etc.~~... The resulting state has energy~~ <~~math>m'+\frac{l^2}{2}</math~~>~~, so the generating function can also be written as~~

	:<~~math~~>~~\left(\sum_{l\in\mathbb{Z}}q^{l^2/2}z^l\right)\left(\sum_{n\geq0}p(n)q^n\right)=\left(\sum_{l\in\mathbb{Z}}q^{l^2/2}z^l\right)\left(\prod_{n>0}(1~~-~~q^n)^{-1}\right)</math>~~

	~~where <math>p(n)</math> is~~ the ~~[[Partition function~~ (~~number theory)\|partition function]].~~ [http://~~arxiv~~.~~org~~/~~abs~~/~~math-ph~~/~~0309015 The uses of random partitions] by [[Andrei Okounkov]] contains a picture of a partition exciting the vacuum.~~

	~~== Notes ==~~
	~~<references/>~~

	~~==References==~~
	* See chapter 14, theorem 14.6 of {{Apostol IANT}}
	* Peter J. Cameron, ''Combinatorics: Topics, Techniques, Algorithms'', (1994) Cambridge University Press, ISBN 0-~~521~~-~~45761~~-0
	*{{Citation \| last1=Jacobi \| first1=C. G. J. \| title=Fundamenta nova theoriae functionum ellipticarum \| url=http://archive.org/details/fundamentanovat00jacogoog \| publisher=Borntraeger\|place=Königsberg \| language=Latin \| isbn=978-1-108-05200-9 \| id=Reprinted by Cambridge University Press 2012 \| year=1829}}

	~~[[Category:Elliptic functions]]~~
	~~[[Category:Theta functions]]~~
	~~[[Category:Mathematical identities]]~~
	~~[[Category:Number theory]~~]

en>Alkauskas: /* References */

2013-11-01T14:59:20Z

References

← Older revision		Revision as of 16:59, 1 November 2013
Line 1:		Line 1:
	~~Nice~~ to ~~meet you, I am Marvella Shryock~~. For a ~~while she~~'~~s been in South Dakota~~. ~~Hiring~~ is ~~his profession~~. ~~What I adore performing~~ is to ~~collect badges but I~~'~~ve been using on new issues recently~~.<br><br>~~My webpage~~ - [http://~~www~~.~~neweracinema~~.~~com~~/~~tube~~/~~user~~/~~GBoykin at home std testing~~]		In [[mathematics]], the '''Jacobi triple product''' is the mathematical identity:

			:<math>\prod_{m=1}^\infty
			\left( 1 - x^{2m}\right)
			\left( 1 + x^{2m-1} y^2\right)
			\left( 1 + x^{2m-1} y^{-2}\right)
			= \sum_{n=-\infty}^\infty x^{n^2} y^{2n},
			</math>
			for complex numbers ''x'' and ''y'', with \|''x''\| < 1 and ''y'' ≠ 0.

			It was introduced by {{harvs\|txt\|authorlink=Carl Gustav Jacob Jacobi\|last=Jacobi\|year=1829}} in his work ''[[Fundamenta Nova Theoriae Functionum Ellipticarum]]''.

			The Jacobi triple product identity is the [[Macdonald identity]] for the affine root system of type ''A''<sub>1</sub>, and is the [[Weyl denominator formula]] for the corresponding affine [[Kac–Moody algebra]].

			== Properties ==

			The basis of Jacobi's proof relies on Euler's [[pentagonal number theorem]], which is itself a specific case of the Jacobi Triple Product Identity.

			Let <math>x=q^{3/2}</math> and <math>y^2=-\sqrt{q}</math>. Then we have
			:<math>\phi(q) = \prod_{m=1}^\infty \left(1-q^m \right) =
			\sum_{n=-\infty}^\infty (-1)^n q^{(3n^2-n)/2}.\, </math>

			The Jacobi Triple Product also allows the Jacobi [[theta function]] to be written as an infinite product as follows:

			Let <math>x=e^{i\pi \tau}</math> and <math>y=e^{i\pi z}.</math>

			Then the Jacobi theta function

			:<math>
			\vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp (\pi i n^2 \tau + 2 \pi i n z)
			</math>

			can be written in the form

			:<math>\sum_{n=-\infty}^\infty y^{2n}x^{n^2}. </math>

			Using the Jacobi Triple Product Identity we can then write the theta function as the product

			:<math>\vartheta(z; \tau) = \prod_{m=1}^\infty
			\left( 1 - \exp(2m \pi i \tau)\right)
			\left( 1 + \exp((2m-1) \pi i \tau + 2 \pi i z)\right)
			\left( 1 + \exp((2m-1) \pi i \tau -2 \pi i z)\right).
			</math>

			There are many different notations used to express the Jacobi triple product. It takes on a concise form when expressed in terms of [[q-Pochhammer symbol]]s:

			:<math>\sum_{n=-\infty}^\infty q^{n(n+1)/2}z^n =
			(q;q)_\infty \; (-1/z;q)_\infty \; (-zq;q)_\infty.</math>

			Where <math>(a;q)_\infty</math> is the infinite ''q''-Pochhammer symbol.

			It enjoys a particularly elegant form when expressed in terms of the [[Ramanujan theta function]]. For <math>\|ab\|<1.</math> it can be written as

			:<math>\sum_{n=-\infty}^\infty a^{n(n+1)/2} \; b^{n(n-1)/2} = (-a; ab)_\infty \;(-b; ab)_\infty \;(ab;ab)_\infty.</math>

			==Proof==
			This proof uses a simplified model of the [[Dirac sea]] and follows the proof in Cameron (13.3) which is attributed to [[Richard Borcherds]]. It treats the case where the power series are formal. For the analytic case, see Apostol. The Jacobi triple product identity can be expressed as

			:<math>\prod_{n>0}(1+q^{n-\frac{1}{2}}z)(1+q^{n-\frac{1}{2}}z^{-1})=\left(\sum_{l\in\mathbb{Z}}q^{l^2/2}z^l\right)\left(\prod_{n>0}(1-q^n)^{-1}\right).</math>

			A ''level'' is a [[half-integer]]. The vacuum state is the set of all negative levels. A state is a set of levels whose symmetric difference with the vacuum state is finite. The ''energy'' of the state <math>S</math> is

			:<math>\sum\{v\colon v > 0,v\in S\} - \sum\{v\colon v < 0, v\not\in S\}</math>

			and the ''particle number'' of <math>S</math> is

			:<math>\|\{v\colon v>0,v\in S\}\|-\|\{v\colon v<0,v\not\in S\}\|.</math>

			An unordered choice of the presence of finitely many positive levels and the absence of finitely many negative levels (relative to the vacuum) corresponds to a state, so the generating function <math>\textstyle\sum_{m,l} s(m,l)q^mz^l</math> for the number <math>s(m,l)</math> of states of energy <math>m</math> with <math>l</math> particles can be expressed as

			:<math>\prod_{n>0}(1+q^{n-\frac{1}{2}}z)(1+q^{n-\frac{1}{2}}z^{-1}).</math>

			On the other hand, any state with <math>l</math> particles can be obtained from the lowest energy <math>l-</math>particle state, <math>\{v\colon v<l\}</math>, by rearranging particles: take a partition <math>\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_j</math> of <math>m'</math> and move the top particle up by <math>\lambda_1</math> levels, the next highest particle up by <math>\lambda_2</math> levels, etc.... The resulting state has energy <math>m'+\frac{l^2}{2}</math>, so the generating function can also be written as

			:<math>\left(\sum_{l\in\mathbb{Z}}q^{l^2/2}z^l\right)\left(\sum_{n\geq0}p(n)q^n\right)=\left(\sum_{l\in\mathbb{Z}}q^{l^2/2}z^l\right)\left(\prod_{n>0}(1-q^n)^{-1}\right)</math>

			where <math>p(n)</math> is the [[Partition function (number theory)\|partition function]]. [http://arxiv.org/abs/math-ph/0309015 The uses of random partitions] by [[Andrei Okounkov]] contains a picture of a partition exciting the vacuum.

			== Notes ==
			<references/>

			==References==
			* See chapter 14, theorem 14.6 of {{Apostol IANT}}
			* Peter J. Cameron, ''Combinatorics: Topics, Techniques, Algorithms'', (1994) Cambridge University Press, ISBN 0-521-45761-0
			*{{Citation \| last1=Jacobi \| first1=C. G. J. \| title=Fundamenta nova theoriae functionum ellipticarum \| url=http://archive.org/details/fundamentanovat00jacogoog \| publisher=Borntraeger\|place=Königsberg \| language=Latin \| isbn=978-1-108-05200-9 \| id=Reprinted by Cambridge University Press 2012 \| year=1829}}

			[[Category:Elliptic functions]]
			[[Category:Theta functions]]
			[[Category:Mathematical identities]]
			[[Category:Number theory]]

en>EmilJ: /* Self-symmetry */ fix link

2012-07-13T17:22:41Z

Self-symmetry: fix link

New page

Nice to meet you, I am Marvella Shryock. For a while she's been in South Dakota. Hiring is his profession. What I adore performing is to collect badges but I've been using on new issues recently.<br><br>My webpage - [http://www.neweracinema.com/tube/user/GBoykin at home std testing]