Questions tagged [theta-functions]

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

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Intuition behind theta function L-series correspondence

I know that we can analytically continue $L$-series by taking the Mellin transform of a sutiable theta function. Technically we need a transformation law for the theta function at $0$ and $\infty$ as ...
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3 votes
2 answers
58 views

A series in terms of the Jacobi theta fuctions

On Wikipedia, one can find an expression of the series $$ \sum_{n=1}^\infty \frac{n^pq^n}{1-q^n} $$ in terms of the Jacobi theta functions for $p=3,5,7$. I'm looking for an expression of this series ...
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6 votes
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Summation in the form of Jacobi Theta Function

TL;DR: My summation should give the same result when it is expressed as the Jacobi Theta function. It gives the same results for some set of inputs but then gives exactly the half of it for other set ...
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Investigating limit of theta like series

Let $v>0$. I want to prove $$\lim_{t \to \infty} \sum_{a \in \mathbb Z} \sum_{b=1}^\infty \exp \left(-v(a/t+bt)^2\right) = 0.$$ It looks quite similar to the theta function $$\theta(z) = \sum_{n \...
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How can one find the zeros of the double theta function? [closed]

What are the zeros of this double theta function? $$ \vartheta=\sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} e^{\theta} $$ $$ \theta={\frac{1}{4} (2 m + \mu)^2 i \pi \tau_{1,1}+\frac{1}{4} (2 n +...
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Quadratic Reciprocity as an Analytic Statement

I was told an interesting fact that quadratic reciprocity follows from the modularity of the theta function $\theta(z) = \sum_{n \in \mathbb{Z}}e^{2\pi in^{2}z}$: $$\theta(\gamma z) = \left(\frac{c}{d}...
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2 votes
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Need help understanding proof of the functional equation for the theta function.

I am following the proof of theorem 1.3 above until at the point the author finds the recurrence relationship $a_n = a_{n-k}e^{b - 2\pi ni\tau}$. When $k = 0$ we have $a_n = a_{n}e^{b - 2\pi ni\tau} \...
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A convergence lemma for adelic zeta function in automorphic forms

I'm reading Godement-Jacquet's classic Zeta functions of simple algebras (1972, Springer). On page 153, the first line: " We also $\textbf{take for granted}$ the following lemma (numbered 11.3)......
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Integral of a sin function over the momentum space resulting in $\delta$ function

I am reading a paper by Faddeev and Kulish, whose name is asymptotic conditions and infrared divergences in QED (the paper is not important for the question I think, but I will include the link: https:...
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In which dimensions do there exist inequivalent lattices with the same theta function?

Equivalently, "Is a lattice determined by the distances (with multiplicities) of its points from the origin?" By a lattice $L$ I mean a discrete additive subgroup of Euclidean space $\mathbb{...
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Verifying Modularity for the Theta Function

I was trying to verify modularity of the theta function $$\theta(z) = \sum_{t \in \mathbb{Z}}e^{2\pi it^{2}z}$$ for $\Gamma_{0}(4)$ directly. I know the factor of automorphy should be $$j(\gamma,z) = \...
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Is there any definition for this series function $f(s)=\sum_{n=1}^\infty e^{-n^s}$?

What I am asking for is if there is any theory related to this real series $f(s)=\sum_{n=1}^\infty e^{-n^s}$ and $s\ge 1$. As far as I know, if $s = 1$, it's a simple geometric series, and if $s = 2$, ...
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Jacobi Elliptic function in terms of theta functions

I am new to using these functions and am confused about what is a function of what. If I want to solve $sn(x,k)$ for a given x, and use this equation: $sn(u,k)=\frac{\theta_{2}}{\theta _{3}}\frac{\...
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5 votes
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141 views

Is this function an alternative solution to the nonlinear pendulum?

Is this function an alternative solution to the nonlinear pendulum? Introduction I am working with the differential equation of the frictionless nonlinear pendulum: $$\ddot{\theta}(t) + b\,\sin(\theta(...
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Gaussian integral using Euler/Jacobi theta function and $r_2(k)$ (number of representations as sum of 2 squares)

The Euler/Jacobi theta function (using the notation of this question) is $\vartheta_3(\tau) := \sum_{n\in \mathbb Z} q^{n^2}$ where $q = e^{2\pi i\tau}$ is the nome. The square $(\vartheta_3(\tau))^2$ ...
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Quintuple product identity in terms of Ramanujan Theta function f(a,b)

The Quintuple product identity is given by $$\sum_{n=-\infty}^{\infty}q^{n(3n+1)/2}\left(z^{-3n}-z^{-3n-1}\right)=(qz:q)_\infty(1/z:q)_\infty(q:q)_\infty(qz^2:q^2)_\infty(q/z^2:q^2)_\infty$$ and This ...
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5 votes
1 answer
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Relation Between Jacobi's Theta Function and Weierstrass $\wp$ Function

I am reading Elliptic Curves by Moll and McKean and it defines Jacobi's theta function on the lattice $\Gamma = \{n+m\omega \mid m,n \in \mathbb{Z}\}$ for a $\omega \in \mathbb{H}$ as below: $$\...
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$\sum_{n=1}^{\infty} \frac{\sin(n^2)}{n^2}$

Question: $$\sum_{n=1}^{\infty}\frac{\sin(n^2)}{n^2}=\,?$$ Previously I calculated a similar summation but it was more luck than wisdom, and insight led me to believe my methods were super incorrect (...
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Approximation for a series involving the derivative of a Jacobi theta function

I’ve considered the diffusion equation $$\frac{\partial f(x,t)}{\partial t}=\frac12 \frac{\partial^2 f(x,t)}{\partial x^2}$$ with the conditions $f(x,0)=\delta(x)$ and $f(-1,t)=f(1,t)=0\ \forall t>...
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1 vote
1 answer
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Can I express this sum as product of two theta functions?

I have an infinite sum written as $$ \sum_{mn} e^{-i2\pi(m c_1 +n c_2)} e^{-(m^2+n^2-mn)} $$ where $m,n$ are integers, $0<c_1, c_2<1$. I want to express the above expression into a product of ...
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Evaluating pattern for summation of Euler-like product

I was inspired by Euler's pentagonal number theorem to play with some products so I began evaluating $$ \prod_{k=1}^{\infty} \left[ 1+\frac{-1+i\sqrt{3}}{2}x^k + \frac{-1-i\sqrt{3}}{2}x^{2k}\right] = \...
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2 votes
0 answers
63 views

A "surprising" asymptotic inverse of $\vartheta _3(0,x)$

After this question of mine related to the problem of approximate solutions of $$\large\color{red}{\operatorname{\vartheta}_3}\left(0,x\right)=k$$ when $k$ is large, continuing the previous work (just ...
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4 votes
1 answer
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Is there an analytic expression for $\sum\limits_{n=0}^{\infty}x^{n^2}$

In statistical mechanics I often come across average energies of the form: $$\begin{equation} \langle\epsilon_n\rangle=\alpha \sum_{n=0}^{\infty}n^2e^{-\alpha n^2} \end{equation}$$ where $\alpha$ is ...
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Connection between elliptic integrals and theta functions

I have read about connections between elliptic integrals and their connections to the Jacobi theta functions, like $\theta_3^2(q) = \frac{2}{\pi}K(k)$, where $q=e^{-\pi\frac{K’(k)}{K(k)}}$, but how ...
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Does this Auxiliary Fresnel Sum=$\frac1{2\sqrt2\pi}\int \limits_0^\infty \frac{\vartheta_3\left(e^{-\frac{\pi x}2}\right)\sqrt x}{x^2+1}dx +\frac14 $?

$$\large{\text{Motivation:}}$$ Here is a related Fresnel Integral sum for a seventh in a series of a sum of just a single function: On $$\mathrm{\sum\limits_{n=0}^\infty \left(C(n)-\frac{\sqrt\pi}{2\...
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2 votes
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Solving for $x$ the equation $\operatorname{\vartheta}_3\left(0,x\right)=k \qquad (k \geq 1)$

Trying to answer this recent question, it reminded me a very old problem we faced almost $50$ years ago. Solve for $x$ the equation $$\color{red} {\operatorname{\vartheta}_3\left(0,x\right)=k} \qquad \...
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4 votes
2 answers
180 views

About the asymptotic behavior of specific Jacobi $\theta$ function $\operatorname{\vartheta}_3\left(0;x\right)$ when $x\to{1-}$.

Since $\displaystyle\sum_{n=1}^\infty{x^{n^2}}=\dfrac{\operatorname{\vartheta}_3\left(0,x\right)-1}2$ for $x\in\left(0,1\right)$ (just in case), it suffices to consider the former below. (Another ...
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1 vote
1 answer
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Closed form for the indefinite integral of the Jacobi Theta function

I am interested in the indefinite integral of $\vartheta_3(q;0)$. A lazy result gives us $$ \int\vartheta_3(q;0)\mathrm{d}q=q+2\sum_{n=1}^\infty \frac{q^{n^2+1}}{n^2+1}+C. $$ While there is nothing ...
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Proof of Jacobi fraction expansion in Triple Product Proof

In the proof of Jacobi's triple product identity by Jacobi, he considers the infinite product; $\frac{1}{(1-qz)(1-q^2z)...}$ and expands it into 1 + $\frac{B_1z}{(1-qz)}$ + $\frac{B_2z^2}{(1-qz)(1-q^...
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0 votes
1 answer
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How to choose witnesses for asymptotic growth?

I'm struggling with how to choose witnesses for asymptotic growth. Specifically, here is the problem I'm working on and what I have done so far: Prove: $$4n^5 – 50n^2 + 10n \in \Theta(n^5)$$ $$0 \leq ...
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1 vote
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closed form in terms of a polygamma function? $ f(2,2)=\sum \sum \exp\big(- n^2k^2 \big)? $

Is there a closed form for:$$ f(2,2)=\sum_{n=1}^\infty \sum_{k=1}^\infty \exp\big(- n^2k^2 \big)? $$ Note that I'm defining a function: $$f(x,y)=\sum_{n=1}^\infty \sum_{k=1}^\infty\exp\big(-n^xk^y\...
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2 votes
1 answer
108 views

What is the asymptotics of: $\Re\left(\frac{\zeta \left(1+\frac{1}{c}\right) \zeta (s+i t)}{\zeta \left(s+i t+\frac{1}{c}+1-1\right)}\right)$?

What is the asymptotics of the function $f(s,t,c)$: $$f(s,t,c)=\Re\left(\frac{\zeta \left(1+\frac{1}{c}\right) \zeta (s+i t)}{\zeta \left(s+i t+\frac{1}{c}+1-1\right)}\right)$$ ? For $c=10^4$ the ...
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2 votes
0 answers
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Weierstrass factorization of $\theta_1(z|\tau)$ in $z$

Suppose $f$ is an entire function and $f(0)\ne 0$. Let $z_1, z_2,\ldots$ be the zeros of $f$ and $p_1, p_2,\ldots$ be any sequence of nonnegative integers such that $$r\gt 0\implies \sum_{n=1}^\infty \...
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5 votes
1 answer
159 views

Closed form of $\sum_{n=1}^\infty \frac{1}{\sinh n\pi}$ in terms of $\Gamma (a)$, $a\in\mathbb{Q}$

This question and this question are about $$\sum_{n=1}^\infty \frac{1}{\cosh n\pi}=\frac{1}{2}\left(\frac{\sqrt{\pi}}{\Gamma ^2(3/4)}-1\right)$$ and $$\sum_{n=1}^\infty \frac{1}{\sinh ^2n\pi}=\frac{1}{...
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2 votes
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Elliptic uniformization of $\sqrt{1+k+k^2}$

I have several elliptic polynomials $P_i(u)$ in Jacobi elliptic functions $sn(u|k)$, $cn(u|k)$ and $dn(u|k)$ with standard definitions $sn^2(u|k)+cn(u|k)^2=1$, $dn^2(u|k)+k^2sn^2(u|k)=1$. Coefficients ...
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2 votes
0 answers
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How to construct the roots of a polynomial via "theta constants"?

I'm wondering about the general solution to the zeroes of polynomials via theta constants, as I came across it the other day on wikipedia. The notation for the theta constants didn't seem too elusive, ...
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4 votes
1 answer
168 views

Explanation of several remarks of Gauss on representations 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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1 vote
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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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6 votes
1 answer
133 views

Is $\prod_{n=0}^\infty \left(1-\frac{1}{\cosh ^2((n+1/2)\pi)}\right)=\frac{1}{\sqrt[4]{2}}$ true?

The infinite product $$\prod_{n=0}^\infty \left(1-\frac{1}{\cosh  ^2((n+1/2)\pi)}\right)$$ agrees with $\frac{1}{\sqrt[4]{2}}$ to at least 100 decimal places. The "identity" is reminiscent ...
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71 views

Absolute value of the theta function on the unit nome

I would like to evalue the absolute value of the theta function with unit nome, i.e. when $|q|=1$ or equivelantly when $\tau\in\mathbb R$. The theta function is expressed as $$\vartheta(z;q=e^{\pi i \...
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  • 411
1 vote
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Jacobi Quintuple Product Identity

I have to show the Jacobi Quintuple Product Identity $$\prod_{n = 1}^{\infty} (1-q^n)(1- \zeta q^{n-1})(1-\zeta^{-1} q^{n})(1-\zeta^{2} q^{2n-1})(1-\zeta^{-2}q^{2n-1}) = \sum_{n \in \mathbb{Z}} q^{\...
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2 votes
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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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  • 509
1 vote
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Proving $\vartheta'_1 (0) = \vartheta_2 (0) \vartheta_3 (0) \vartheta_4 (0)$ different from Whittaker and Watson

So I am self-learning about Jacobi theta function from Whittaker and Watson's book. In section $\textbf{21.41}$, they introduce the proof for the identity below: $$\vartheta'_1 (0) = \vartheta_2 (0) \...
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5 votes
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174 views

Validity of $\ln z=\frac{\pi}{\operatorname{AGP}(\theta_2^2(1/z),\theta_3^2(1/z))}$

Definitions For the definitions of $\operatorname{AGP}$ and $\operatorname{AGO}$, see here or here. $\theta_2(z)$ and $\theta_3(z)$ are defined as follows: $$\theta_2(z)=\sum_{n=-\infty}^\infty z^{(n+...
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4 votes
0 answers
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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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4 votes
2 answers
92 views

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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0 votes
1 answer
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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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100 views

Prove that $\sum _{n=-\infty }^{\infty } e^{-n^2 \pi x}=\frac{1}{\sqrt{x}}\sum _{n=-\infty }^{\infty } e^{-\frac{n^2 \pi }{x}}$ [duplicate]

In Wikipedia's proof of Riemann's functional equation for the zeta function (here, and click "Show Proof"), I find the assertion that $$\sum _{n=-\infty }^{\infty } e^{-n^2 \pi x} = \frac{1}{...
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128 views

Square of Jacobi theta function is sum of hyperbolic secant?

I'm presently reading Henri Cohen's Introduction to Modular Forms (https://arxiv.org/pdf/1809.10907.pdf) and I'm trying to do exercise 1.5, which partially entails showing that: $T_2(a)\equiv\sum_{n=-\...
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1 vote
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68 views

Symbolic calculation vs. desmos

Context: I am working on getting a recurrence of the form $$nc_n=\sum_{k=1}^nc_{n-k}R(k)\tag0$$ for the coefficients $c_n$ defined by $$f(q)=\vartheta_3^2(q)=\prod_{m\ge1}(1+q^{2m-1})^4(1-q^{2m})^2=\...
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