Ramanujan theta function: Difference between revisions
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{{Short description|Mathematical function}} |
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{{Distinguish|text=the [[mock theta function]]s discovered by Ramanujan}} |
{{Distinguish|text=the [[mock theta function]]s discovered by Ramanujan}} |
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In [[mathematics]], particularly [[q-analog]] theory, the '''Ramanujan theta function''' generalizes the form of the Jacobi [[theta function]]s, while capturing their general properties. In particular, the [[Jacobi triple product]] takes on a particularly elegant form when written in terms of the Ramanujan theta. The function is named after [[Srinivasa Ramanujan]] |
In [[mathematics]], particularly [[q-analog|{{mvar|q}}-analog]] theory, the '''Ramanujan theta function''' generalizes the form of the Jacobi [[theta function]]s, while capturing their general properties. In particular, the [[Jacobi triple product]] takes on a particularly elegant form when written in terms of the Ramanujan theta. The function is named after mathematician [[Srinivasa Ramanujan]]. |
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==Definition== |
==Definition== |
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:<math>f(a,b) = \sum_{n=-\infty}^\infty |
:<math>f(a,b) = \sum_{n=-\infty}^\infty |
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a^{n(n+1) |
a^\frac{n(n+1)}{2} \; b^\frac{n(n-1)}{2} </math> |
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for |''ab'' |
for {{math|{{abs|''ab''}} < 1}}. The [[Jacobi triple product]] identity then takes the form |
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:<math>f(a,b) = (-a; ab)_\infty \;(-b; ab)_\infty \;(ab;ab)_\infty.</math> |
:<math>f(a,b) = (-a; ab)_\infty \;(-b; ab)_\infty \;(ab;ab)_\infty.</math> |
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Here, the expression <math>(a;q)_n</math> denotes the [[q-Pochhammer symbol]]. Identities that follow from this include |
Here, the expression <math>(a;q)_n</math> denotes the [[q-Pochhammer symbol|{{mvar|q}}-Pochhammer symbol]]. Identities that follow from this include |
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:<math>\varphi(q) = f(q,q) = \sum_{n=-\infty}^\infty q^{n^2} = |
:<math>\varphi(q) = f(q,q) = \sum_{n=-\infty}^\infty q^{n^2} = |
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{(-q;q^2)_\infty^2 (q^2;q^2)_\infty} </math> |
{\left(-q;q^2\right)_\infty^2 \left(q^2;q^2\right)_\infty} </math> |
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and |
and |
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:<math>\psi(q) = f(q,q^3) = \sum_{n=0}^\infty q^{n(n+1) |
:<math>\psi(q) = f\left(q,q^3\right) = \sum_{n=0}^\infty q^\frac{n(n+1)}{2} = |
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{(q^2;q^2)_\infty}{(-q; q)_\infty} </math> |
{\left(q^2;q^2\right)_\infty}{(-q; q)_\infty} </math> |
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and |
and |
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:<math>f(-q) = f(-q,-q^2) = \sum_{n=-\infty}^\infty (-1)^n q^{n(3n-1) |
:<math>f(-q) = f\left(-q,-q^2\right) = \sum_{n=-\infty}^\infty (-1)^n q^\frac{n(3n-1)}{2} = |
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(q;q)_\infty </math> |
(q;q)_\infty </math> |
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This last being the [[Euler function]], which is closely related to the [[Dedekind eta function]]. The Jacobi [[theta function]] may be written in terms of the Ramanujan theta function as: |
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:<math>\vartheta(w, q)=f(qw^2,qw^{-2})</math> |
:<math>\vartheta(w, q)=f\left(qw^2,qw^{-2}\right)</math> |
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==Integral representations== |
==Integral representations== |
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We have the following integral representation for the full two-parameter form of Ramanujan's theta function:<ref name="SQSERIES-MDS">{{cite journal|last1=Schmidt|first1=M. D.|title=Square series generating function transformations|journal=Journal of Inequalities and Special Functions|date=2017|volume=8|issue=2|url=http://www.ilirias.com/jiasf/repository/docs/JIASF8-2-11.pdf}}</ref> |
We have the following integral representation for the full two-parameter form of Ramanujan's theta function:<ref name="SQSERIES-MDS">{{cite journal|last1=Schmidt|first1=M. D.|title=Square series generating function transformations|journal=Journal of Inequalities and Special Functions|date=2017|volume=8|issue=2|arxiv=1609.02803|url=http://www.ilirias.com/jiasf/repository/docs/JIASF8-2-11.pdf}}</ref> |
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:<math> |
:<math> |
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f(a,b) = 1 + \int_0^{\infty} \frac{2a e^{-\frac12 t^2}}{\sqrt{2\pi}}\left[ |
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\begin{align} |
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\frac{1 - a \sqrt{ab} \cosh\left(\sqrt{\log ab} \,t\right)}{ |
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a^3 b - 2a \sqrt{ab} \cosh\left(\sqrt{\log ab} \,t\right) + 1} |
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a^3 b - 2a \sqrt{ab} \cosh\left(\sqrt{\log(ab)} t\right) + 1} |
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\right] dt + |
\right] dt + |
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\int_0^{\infty} \frac{2b e^{-t^ |
\int_0^{\infty} \frac{2b e^{-\frac12 t^2}}{\sqrt{2\pi}}\left[ |
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\frac{1 - b \sqrt{ab} \cosh\left(\sqrt{\log |
\frac{1 - b \sqrt{ab} \cosh\left(\sqrt{\log ab} \,t\right)}{ |
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a b^3 - 2b \sqrt{ab} \cosh\left(\sqrt{\log |
a b^3 - 2b \sqrt{ab} \cosh\left(\sqrt{\log ab} \,t\right) + 1} |
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\right] dt |
\right] dt |
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\end{align} |
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</math> |
</math> |
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The special cases of Ramanujan's theta functions given by |
The special cases of Ramanujan's theta functions given by {{math|''φ''(''q'') :{{=}} ''f''(''q'', ''q'')}} {{oeis|id=A000122}} and {{math|''ψ''(''q'') :{{=}} ''f''(''q'', ''q''<sup>3</sup>)}} {{oeis|id=A010054}} <ref>{{cite web|last1=Weisstein|first1=Eric W.|title=Ramanujan Theta Functions|url=http://mathworld.wolfram.com/RamanujanThetaFunctions.html|website=MathWorld|accessdate=29 April 2018}}</ref> also have the following integral representations:<ref name="SQSERIES-MDS" /> |
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:<math> |
:<math> |
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\begin{align} |
\begin{align} |
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\varphi(q) & = 1 + \int_0^{\infty} \frac{e^{-t^ |
\varphi(q) & = 1 + \int_0^{\infty} \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} |
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\left[\frac{4q \left(1-q^2 \cosh\left( |
\left[\frac{4q \left(1-q^2 \cosh\left( |
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\sqrt{2 \log |
\sqrt{2 \log q} \,t\right)\right)}{q^4-2 q^2 |
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\cosh\left(\sqrt{2 \log |
\cosh\left(\sqrt{2 \log q} \,t\right) + 1} |
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\right] dt \\ |
\right] dt \\[6pt] |
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\psi(q) & = \int_0^{\infty} |
\psi(q) & = \int_0^{\infty} |
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\frac{2 e^{-t^ |
\frac{2 e^{-\frac12 t^2}}{\sqrt{2\pi}} |
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\left[\frac{ |
\left[\frac{1-\sqrt{q} |
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\cosh\left(\sqrt{\log |
\cosh\left(\sqrt{\log q} \,t\right)}{q-2 \sqrt{q} |
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\cosh\left(\sqrt{\log |
\cosh\left(\sqrt{\log q} \,t\right) + 1} |
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\right] dt |
\right] dt |
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\end{align} |
\end{align} |
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</math> |
</math> |
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This leads to several special case integrals for constants defined by these functions when |
This leads to several special case integrals for constants defined by these functions when {{math|''q'' :{{=}} ''e''<sup>−''kπ''</sup>}} (cf. [[Theta function#Explicit values|theta function explicit values]]). In particular, we have that <ref name="SQSERIES-MDS" /> |
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:<math> |
:<math> |
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\begin{align} |
\begin{align} |
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\varphi\left(e^{-k\pi}\right) & = |
\varphi\left(e^{-k\pi}\right) & = |
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1 + \int_0^ |
1 + \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ |
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\frac{4 e^{k\pi} \left(e^{2k\pi} - \cos\left(\sqrt{2\pi k} t\right) |
\frac{4 e^{k\pi} \left(e^{2k\pi} - \cos\left(\sqrt{2\pi k} \,t\right) |
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\right)}{e^{4k\pi} - 2 e^{2k\pi} \cos\left(\sqrt{2\pi k} t\right) + 1} |
\right)}{e^{4k\pi} - 2 e^{2k\pi} \cos\left(\sqrt{2\pi k} \,t\right) + 1} |
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\right] dt \\ |
\right] dt \\[6pt] |
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\frac{\pi^ |
\frac{\pi^\frac14}{\Gamma\left(\frac34\right)} & = |
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1 + \int_0^ |
1 + \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ |
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\frac{4 e^ |
\frac{4 e^\pi \left(e^{2\pi} - \cos\left(\sqrt{2\pi} \,t\right) |
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\right)}{e^{4\pi} - 2 e^{2\pi} \cos\left(\sqrt{2\pi} t\right) + 1} |
\right)}{e^{4\pi} - 2 e^{2\pi} \cos\left(\sqrt{2\pi} \,t\right) + 1} |
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\right] dt \\ |
\right] dt \\[6pt] |
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\frac{\pi^ |
\frac{\pi^\frac14}{\Gamma\left(\frac34\right)} \cdot |
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\frac |
\frac{\sqrt{2 + \sqrt{2}}}{2} & = |
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1 + \int_0^ |
1 + \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ |
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\frac{4 e^{2\pi} \left(e^{4\pi} - \cos\left(2 \sqrt{\pi} t\right) |
\frac{4 e^{2\pi} \left(e^{4\pi} - \cos\left(2 \sqrt{\pi} \,t\right) |
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\right)}{e^{8\pi} - 2 e^{4\pi} \cos\left(2 \sqrt{\pi} t\right) + 1} |
\right)}{e^{8\pi} - 2 e^{4\pi} \cos\left(2 \sqrt{\pi} \,t\right) + 1} |
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\right] dt \\ |
\right] dt \\[6pt] |
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\frac{\pi^ |
\frac{\pi^\frac14}{\Gamma\left(\frac34\right)} \cdot |
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\frac{\sqrt{\sqrt{3} |
\frac{\sqrt{1 + \sqrt{3}}}{2^\frac14 3^\frac38} & = |
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1 + \int_0^ |
1 + \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ |
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\frac{4 e^{3\pi} \left(e^{6\pi} - \cos\left(\sqrt{6 \pi} t\right) |
\frac{4 e^{3\pi} \left(e^{6\pi} - \cos\left(\sqrt{6 \pi} \,t\right) |
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\right)}{e^{12\pi} - 2 e^{6\pi} \cos\left(\sqrt{6 \pi} t\right) + 1} |
\right)}{e^{12\pi} - 2 e^{6\pi} \cos\left(\sqrt{6 \pi} \,t\right) + 1} |
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\right] dt \\ |
\right] dt \\[6pt] |
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\frac{\pi^ |
\frac{\pi^\frac14}{\Gamma\left(\frac34\right)} \cdot |
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\frac{\sqrt{5 + 2 \sqrt{5}}}{5^ |
\frac{\sqrt{5 + 2 \sqrt{5}}}{5^\frac34} & = |
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1 + \int_0^ |
1 + \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ |
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\frac{4 e^{5\pi} \left(e^{10\pi} - \cos\left(\sqrt{10 \pi} t\right) |
\frac{4 e^{5\pi} \left(e^{10\pi} - \cos\left(\sqrt{10 \pi} \,t\right) |
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\right)}{e^{20\pi} - 2 e^{10\pi} \cos\left(\sqrt{10 \pi} t\right) + 1} |
\right)}{e^{20\pi} - 2 e^{10\pi} \cos\left(\sqrt{10 \pi} \,t\right) + 1} |
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\right] dt |
\right] dt |
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\end{align} |
\end{align} |
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</math> |
</math> |
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\begin{align} |
\begin{align} |
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\psi\left(e^{-k\pi}\right) & = |
\psi\left(e^{-k\pi}\right) & = |
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\int_0^ |
\int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ |
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\frac{\cos\left(\sqrt{k \pi} t\right) - e^{k\pi |
\frac{\cos\left(\sqrt{k \pi} \,t\right) - e^\frac{k\pi}{2}}{ |
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\cos\left(\sqrt{k \pi} t\right) - \cosh |
\cos\left(\sqrt{k \pi} \,t\right) - \cosh\frac{k\pi}{2}} |
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\right] dt \\ |
\right] dt \\[6pt] |
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\frac{\pi^ |
\frac{\pi^\frac14}{\Gamma\left(\frac34\right)} \cdot |
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\frac{e^{\pi |
\frac{e^\frac{\pi}{8}}{2^\frac58} & = |
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\int_0^ |
\int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ |
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\frac{\cos\left(\sqrt{\pi} t\right) - e^{\pi |
\frac{\cos\left(\sqrt{\pi} \,t\right) - e^\frac{\pi}{2}}{ |
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\cos\left(\sqrt{\pi} t\right) - \cosh |
\cos\left(\sqrt{\pi} \,t\right) - \cosh\frac{\pi}{2}} |
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\right] dt \\ |
\right] dt \\[6pt] |
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\frac{\pi^ |
\frac{\pi^\frac14}{\Gamma\left(\frac34\right)} \cdot |
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\frac{e^{\pi |
\frac{e^\frac{\pi}{4}}{2^\frac54} & = |
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\int_0^ |
\int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ |
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\frac{\cos\left(\sqrt{2 \pi} t\right) - e^ |
\frac{\cos\left(\sqrt{2 \pi} \,t\right) - e^\pi}{ |
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\cos\left(\sqrt{2 \pi} t\right) - \cosh |
\cos\left(\sqrt{2 \pi} \,t\right) - \cosh \pi} |
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\right] dt \\ |
\right] dt \\[6pt] |
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\frac{\pi^ |
\frac{\pi^\frac14}{\Gamma\left(\frac34\right)} \cdot |
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\frac{ |
\frac{\sqrt[4]{1 + \sqrt{2}} \, e^\frac{\pi}{16}}{2^\frac{7}{16}} & = |
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\int_0^ |
\int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ |
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\frac{\cos\left(\sqrt{\frac{\pi}{2}} t\right) - e^{\pi |
\frac{\cos\left(\sqrt{\frac{\pi}{2}} \,t\right) - e^\frac{\pi}{4}}{ |
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\cos\left(\sqrt{\frac{\pi}{2}} t\right) - \cosh |
\cos\left(\sqrt{\frac{\pi}{2}} \,t\right) - \cosh\frac{\pi}{4}} |
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\right] dt |
\right] dt |
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\end{align} |
\end{align} |
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</math> |
</math> |
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==Application in string theory== |
==Application in string theory== |
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The Ramanujan theta function is used to determine the [[critical dimension]]s in [[ |
The Ramanujan theta function is used to determine the [[critical dimension]]s in [[bosonic string theory]], [[superstring theory]] and [[M-theory]]. |
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==References== |
==References== |
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{{Reflist}} |
{{Reflist}} |
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* {{cite book |first=W. N. |last=Bailey |title=Generalized Hypergeometric Series |year=1935 |series=Cambridge Tracts in Mathematics and Mathematical Physics |volume=32 |publisher=Cambridge University Press |location=Cambridge }} |
* {{cite book |first=W. N. |last=Bailey |title=Generalized Hypergeometric Series |year=1935 |series=Cambridge Tracts in Mathematics and Mathematical Physics |volume=32 |publisher=Cambridge University Press |location=Cambridge }} |
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* {{cite book | |
* {{cite book |first1=George |last1=Gasper |first2=Mizan |last2=Rahman |title=Basic Hypergeometric Series |edition=2nd |year=2004 |series=Encyclopedia of Mathematics and Its Applications |volume=96 |publisher=Cambridge University Press |location=Cambridge |isbn=0-521-83357-4 }} |
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* {{springer|title=Ramanujan function|id=p/r077200}} |
* {{springer|title=Ramanujan function|id=p/r077200}} |
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* {{cite book |last=Kaku |first=Michio |year=1994 |title=Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension |location=Oxford |publisher=Oxford University Press |isbn=0-19-286189-1 }} |
* {{cite book |last=Kaku |first=Michio |year=1994 |title=Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension |location=Oxford |publisher=Oxford University Press |isbn=0-19-286189-1 }} |
Latest revision as of 23:03, 22 March 2024
In mathematics, particularly q-analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi triple product takes on a particularly elegant form when written in terms of the Ramanujan theta. The function is named after mathematician Srinivasa Ramanujan.
Definition
[edit]The Ramanujan theta function is defined as
for |ab| < 1. The Jacobi triple product identity then takes the form
Here, the expression denotes the q-Pochhammer symbol. Identities that follow from this include
and
and
This last being the Euler function, which is closely related to the Dedekind eta function. The Jacobi theta function may be written in terms of the Ramanujan theta function as:
Integral representations
[edit]We have the following integral representation for the full two-parameter form of Ramanujan's theta function:[1]
The special cases of Ramanujan's theta functions given by φ(q) := f(q, q) OEIS: A000122 and ψ(q) := f(q, q3) OEIS: A010054 [2] also have the following integral representations:[1]
This leads to several special case integrals for constants defined by these functions when q := e−kπ (cf. theta function explicit values). In particular, we have that [1]
and that
Application in string theory
[edit]The Ramanujan theta function is used to determine the critical dimensions in bosonic string theory, superstring theory and M-theory.
References
[edit]- ^ a b c Schmidt, M. D. (2017). "Square series generating function transformations" (PDF). Journal of Inequalities and Special Functions. 8 (2). arXiv:1609.02803.
- ^ Weisstein, Eric W. "Ramanujan Theta Functions". MathWorld. Retrieved 29 April 2018.
- Bailey, W. N. (1935). Generalized Hypergeometric Series. Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 32. Cambridge: Cambridge University Press.
- Gasper, George; Rahman, Mizan (2004). Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications. Vol. 96 (2nd ed.). Cambridge: Cambridge University Press. ISBN 0-521-83357-4.
- "Ramanujan function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Kaku, Michio (1994). Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension. Oxford: Oxford University Press. ISBN 0-19-286189-1.
- Weisstein, Eric W. "Ramanujan Theta Functions". MathWorld.