# Questions tagged [theta-functions]

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

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### Explanation of several remarks of Gauss on represesentations of a given number as sum of four squares.

Remark 1: On p.384 of volume 3 of Gauss's Werke, which is a part of an unpublished treatise on the arithmetic geometric mean, Gauss makes the following remark: On the theory of the division of ...
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### Uniform convergence of theta functions

I refer to the following lecture notes on theta functions: https://web.math.princeton.edu/~gunning/theta2/A On page 6 Section 2 of these notes (page 13 in the pdf), the definition of a theta function ...
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### Closed forms for these Fourier series?

I recently encountered the following series $$\sum_{m \ne 0, m \in \mathbb{Z}} \frac{1}{(\sin m \pi \tau)^n} e^{2\pi i m z} \ , \quad n= 1, 2, ...$$ For $n = 1, 2$, I managed to find the closed ...
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### Is $\prod_{n=0}^\infty \left(1-\frac{1}{\cosh ^2((n+1/2)\pi)}\right)=\frac{1}{\sqrt{2}}$ true?

The infinite product $$\prod_{n=0}^\infty \left(1-\frac{1}{\cosh ^2((n+1/2)\pi)}\right)$$ agrees with $\frac{1}{\sqrt{2}}$ to at least 100 decimal places. The "identity" is reminiscent ...
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### Prove by induction $\sum_{i=1}^ni^2=1^2+2^2+3^2+…+n^2=\frac{n(n+1)(2n+1)}{6}=Θ(n^3)$ [duplicate]

I am trying to prove $$\sum_{i=1}^ni^2=1^2+2^2+3^2+…+n^2=\frac{n(n+1)(2n+1)}{6}=Θ(n^3)$$ I understand when it comes to big-theta that f(n)= $$\sum_{i=1}^ni^2=1^2+2^2+3^2+…+n^2=\frac{n(n+1)(2n+1)}{6}$$ ...
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### Definite integral involving Jacobi theta function

Consider an elliptic function $f(z)$, we know how to compute the integral $$\int_0^1 f(z)dz,$$ by expanding $f(z)$ as a sum of $\zeta(z)$ and its derivatives, where the coefficients of the expansion ...
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### What is the coefficient of $q^n$ in the Lambert Series Expansion $a^2(q).a(q^5)$

What is the coefficient of $q^n$ in the Lambert Series Expansion $a^2(q).a(q^5)$. Here $a(q)$ is Borwein Theta Function. I am using Ramanujan's Theta Functions Book by Shaun Cooper as a reference ...
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### What does $\sum_{n=0}^\infty z^{n(n+1)/2}$, $|z|<1$ converge to?

Does anyone know what the series $$S(z) = \sum_{n=0}^{\infty} z^\frac{n(n+1)}{2}$$ converges to for $|z|<1$? This came up in an application where $z$ is a probability and $S(z)$ an expected value....
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### What is the connection between different definitions for theta functions?

In his 2002 thesis, Zwegers defines theta functions associated with definite and indefinite quadratic forms. For example, if $Q:\mathbb{R}^r\to \mathbb{R}$ is a positive definite quadratic form with ...
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### Resources for learning about theta functions

I would like to learn about the Jacobi theta functions,however, I'm struggling to find a free ebook /website that would introduce the functions in a detailed comprehensible manner for beginners. So is ...
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### evaluation of $\sum_{n=-\infty}^{\infty}e^{-2\pi^{2}z^{2}n^{2}}$ for large z

I want to evaluate the following \begin{equation} \sum_{n=-\infty}^{\infty}e^{-2\pi^{2}z^{2}n^{2}} \end{equation} I know for $z\ll 1$ we can use Euler-Maclaurin formula but in my case z is quite ...
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### Evaluation of $\sum_{n=1}^{\infty}q^{n^{2}}$?

I would like to evaluate the following summation \begin{equation} \sum_{n=1}^{\infty}q^{n^{2}} \end{equation} assuming $0<q<1$ obviously the series converge but can anyone help me how to ...
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### Finding roots of $\sum\limits_{n = - \infty }^ \infty n z^n q^{n^2} =0$ , $z_k=u_k(q)$

The Jacobi triple product identity is: $$F(z,q)=\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2}$$ where $|q|<1$ All roots ...
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