Questions tagged [theta-functions]

For questions about $\theta$ functions (special functions of several complex variables).

In mathematics, $\theta$ functions are special functions of several complex variables. They are important in many areas, including the theories of abelian varieties and moduli spaces, and of quadratic forms. They have also been applied to soliton theory. When generalized to a Grassmann algebra, they also appear in quantum field theory. The most common form of $\theta$ function is that occurring in the theory of elliptic functions.

284 questions
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Family of definite integrals involving Dedekind eta function of a complex argument, $\int_0^{\infty} \eta^k(ix)dx$

The Dedekind eta function is denoted by $\eta(\tau)$, and is defined on the upper half-plane ($\Im \tau >0$). Put $\tau = i x$ where $x$ is a positive real number. The function has the following representations: $$\eta(ix)= e^{-\pi x/12}…
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Proving $\left(\sum_{n=-\infty}^\infty q^{n^2} \right)^2 = \sum_{n=-\infty}^\infty \frac{1}{\cos(n \pi \tau)}$

The so-called "two squares theorem" can be proven by establishing the following identity: $$\left(\sum_{n=-\infty}^\infty e^{\pi i \tau n^2}\right)^2 = \sum_{n=-\infty}^\infty \frac{1}{\cos(n \pi \tau)}$$ where $\Im \tau>0$. Stein and Shakarchi give…
Argon
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A problem posed by Ramanujan involving $\sum e^{-5\pi n^2}$

While going through the list of problems posed by Ramanujan in Journal of Indian Mathematical Society I came across this problem involving theta functions: Prove that $$\frac{1}{2}+\sum_{n=1}^{\infty} e^{-\pi n^2x}\cos(\pi…
Paramanand Singh
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What is a Theta Function?

What exactly is a theta function $$\vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp (\pi i n^2 \tau + 2 \pi i n z)= 1 + 2 \sum_{n=1}^\infty \left(e^{\pi i\tau}\right)^{n^2} \cos(2\pi n z) = \sum_{n=-\infty}^\infty q^{n^2}\eta^n"$$ and how does it…
bolbteppa
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Convexity of $\theta(q)$

Define Jacobi's (fourth) theta function with argument zero and nome $q$: $$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$ plot of the function via Wolfram|Alpha plot of the function via Sage I am looking for a simple/standard/illuminating proof…
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Asymptotic equivalent of $\sum_{n\ge0} q^{n^2}{x^n}$ as $x\to+\infty$

Let $q\in\Bbb C^*$ with $|q|<1$, define $$f:x\mapsto\sum_{n\ge0} q^{n^2}{x^n}$$ I want to find an asymptotic equivalent of $f$ as $x\to+\infty$. I found that $$a\le|f(x)|\cdot\exp\left(\frac{\ln^2|x|}{4\ln |q|}\right)\le b$$ where …
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Hermite's solution of the general quintic in terms of theta functions

Can someone point me at or produce a translation or modern exposition of Hermite's solution of the general quintic in terms of theta functions? (the "before" and "after" steps are on the mathworld page for the quintic, but I'm interested in…
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An interesting identity involving Jacobi $\theta_4$ and $\zeta(2)$

A recent question mentioned an integral identity involving Dedekind $\eta$ function and a special value for the complete elliptic integral of the first kind. I refrained from providing a complete answer, I rather tried to guide the OP through some…
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Combinatorial interpretation of this identity of Gauss?

Gauss came up with some bizarre identities, namely $$ \sum_{n\in\mathbb{Z}}(-1)^nq^{n^2}=\prod_{k\geq 1}\frac{1-q^k}{1+q^k}. $$ How can this be interpreted combinatorially? It strikes me as being similar to many partition identities. Thanks.
Kally
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series involving $\log \left(\tanh\frac{\pi k}{2} \right)$

I found an interesting series $$\sum_{k=1}^\infty \log \left(\tanh \frac{\pi k}{2} \right)=\log(\vartheta_4(e^{-\pi}))=\log \left(\frac{\pi^{\frac{1}{4}}}{2^{\frac{1}{4}}\Gamma \left( \frac{3}{4}\right)} \right)$$ Does anybody know how to approach…
14
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Closed-form of an integral involving a Jacobi theta function, $ \int_0^{\infty} \frac{\theta_4^{n}\left(e^{-\pi x}\right)}{1+x^2} dx $

The Jacobi theta function $\theta_4$ is defined by $$\displaystyle \theta_4(q)=\sum_{n \in \mathbb{Z}} (-1)^n q^{n^2} \tag{1}$$ For this question, set $q=\large e^{-\pi x}$ and $\theta_4 \equiv \theta_4(q)$. Define $\theta_3(q)=\theta_4(-q)$. Using…
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Summation of $\sum_{n=0}^{\infty}a^nq^{n^2}$

I am trying to find the result for the sum of the form $\sum_{n=0}^{\infty}a^nq^{n^2}$. The special case for $a=1$ is easily given by $\vartheta(0,q)$, where $\vartheta(z,q)$ is the third Jacobi Theta function. So, whatever the answer is, it must…
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Extension of the Jacobi triple product identity

The Jacobi triple product identity is: $$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} $$ I would like to extend the idea for $\sum\limits_{n = - \infty }^ \infty z^n q^{n^2}…
Mathlover
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Show that $\prod\limits_{n=1}^\infty \frac{(1-q^{6n})(1-q^n)^2}{(1-q^{3n})(1-q^{2n})}=\sum\limits_{n=-\infty}^\infty q^{2n^2+n}-3q^{9(2n^2+n)+1}$.

Show that $\displaystyle \prod_{n=1}^\infty \frac{(1-q^{6n})(1-q^n)^2}{(1-q^{3n})(1-q^{2n})}=\sum_{n=-\infty}^\infty q^{2n^2+n}-3q^{9(2n^2+n)+1}$. I can't seem to be able to proceed with this question. I know that $\displaystyle \prod_{n=1}^\infty…
Icycarus
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Ramanujan theta function and its continued fraction

I believe Ramanujan would have loved this kind of identity. After deriving the identity, I wanted to share it with the mathematical community. If it's well known, please inform me and give me some links to it. Let $q=e^{2\pi\mathrm{i}\tau}$,…
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