Questions tagged [theta-functions]

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

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0answers
86 views

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Remark 1: On p.384 of volume 3 of Gauss's Werke, which is a part of an unpublished treatise on the arithmetic geometric mean, Gauss makes the following remark: On the theory of the division of ...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

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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34 views

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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1answer
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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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1answer
46 views

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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30 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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31 views

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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46 views

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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34 views

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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159 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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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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2answers
80 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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1answer
42 views

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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69 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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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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51 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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1answer
62 views

Finding $\alpha$ such that $\alpha(f(x)-\sqrt{\pi})=\cos(2\pi x)$, where $f(x)=\sum_{n\in\mathbb{Z}}{e^{-(x-n)^2}}$

Given the function $$f(x)=\sum_{n\in\mathbb{Z}}{e^{-(x-n)^2}}$$ find the constant $\alpha$ (independent of $x$) such that $$\alpha(f(x)-\sqrt{\pi})=\cos(2\pi x)$$ Numerically alpha seems to be around $...
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2answers
153 views

Equivalency between a “mixed modular equation” of Gauss and a later theorem of Ramanujan.

In p. 476 of volume 3 of Gauss's collected works, appear several interesting identities on Jacobi theta functions which were used in 1904 by the czech mathematician Karel Petr to derive relations ...
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2answers
169 views

Interpretation of a certain general theorem used by Gauss in his work on theta functions.

I'm trying to understand the meaning of a general proposition stated by Gauss in a posthomous paper (this paper is in pp. 470-481 of volume 3 of Gauss's werke) on theta functions, a proposition which ...
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81 views

Has there been any exploration on Cubic Power Series?

I was interested in finding some identities/special values involving the function $$\gamma(z) = \sum_{i=0}^{\infty} z^{i^3} = 1 + z + z^8 + z^{27} + ... $$ which can be thought of as a "cubic ...
3
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2answers
78 views

Find an explicit formula for $ \sum_{n=0}^{\infty}{x^{n^2}} ,\quad \forall x \in (0,1) $

My question askes me to find an explicit formula for $$ \sum_{n=0}^{\infty}{x^{n^2}} \quad\left(\forall x \in (0,1)\right)$$ And I feel it kind of interesting to find an appropriate f(x) that ...
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12 views

Transforming computable numbers into analytic functions using diophantine equations?

By Matiyasevich's theorem, any enumerable set $\mathbf{S}$ can be expressed as solutions of a diophantine equation in the integers. These can always be expressed as degree 4 polynomials in the ...
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1answer
113 views

How to prove this equality of series

While studying some physics problems, I stumbled upon this experimental equality: $$ \sum_{k, \ell = 0}^{+\infty} q^{\frac{ 1 }{ 2 }[( k + \ell + 1)^2 - (k- \ell)]} = \frac{ \sqrt{q} }{ 1-q } \ . $$ I ...
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44 views

Identity of modified Bessel functions

Is there something similar to Jacobi's anger expansion for powers of modified Bessel functions? What I mean is an evaluation of the series $$ \sum_{n\in\mathbb{Z}}\mathrm{e}^{\mathrm{i}n\theta}\left(...
4
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1answer
107 views

Logarithmic differentiation of a complicated product

In this answer, it says that one may take the logarithmic derivative w.r.t. $z$ on both sides of the equation $$(q^4;q^4)_\infty\left\{z(-z^4q^3;q^4)_\infty(-z^{-4}q;q^4)_\infty-z^{-1}(-z^4q;q^4)_\...
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1answer
163 views

$\int_0^{\infty}e^{-t^2/2}\,\frac{e^{2\pi}-\cos\left(\sqrt{2\pi} t\right)}{e^{4\pi}-2e^{2\pi}\cos\left(\sqrt{2\pi} t\right)+1} dt $

How does one show $$ \int_0^{\infty} e^{-t^2/2} \left[ \frac{e^{2\pi} - \cos\left(\sqrt{2\pi} t \right)}{e^{4\pi} - 2 e^{2\pi} \cos\left(\sqrt{2\pi} t\right) + 1} \right] dt = \...
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2answers
101 views

Integral with theta function

$$I=\int_{-L/2}^{L/2}dx \Theta(\epsilon-2A|x|)$$ $L>0; A>0$ and $\epsilon$ are parameters, I should solve this integral but I don't know how I thought about solving it like this: $$\epsilon-2A|...
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0answers
128 views

Abel's summation formula and approximating an integral of Jacobi theta functions

As my previous post remains unanswered, I thought I would post a more complete form of the problem in case it would be more practical to work on/ solve. I am trying to compute the following integral \...
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2answers
54 views

Real roots of the equation $2x^2+(4\sin\theta)x+\cos(2\theta)=0$

For what values of $\theta, 0 ≤ \theta ≤ 2π$ does the equation $$2x^2+(4\sin\theta)x+\cos(2\theta)=0$$ have real roots? Also, am I supposed to find specific values of $\theta$ or find an equation that ...
4
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1answer
41 views

How to prove that this is a zero of the theta function?

We have the definition $$\vartheta(\tau, z) = \sum_{n=-\infty}^\infty e^{\pi i \cdot(n^2 \tau + 2 n z)}$$ and I want to show $\vartheta(\tau, \tfrac{\tau + 1}{2}) = 0$. Substituting it in I get $$\...
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2answers
298 views

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 ...
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1answer
103 views

What are specific proofs of Jacobi Triple Product Identity?

I am looking for the Special Proofs. Here is a reference from MSE. Motivation for/history of Jacobi's triple product identity I also know that a simple proof via Functional Equation from the book ...
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43 views

Combined Integral transformations (Mellin + Laplace)

I'm looking for the solution of the following integral $$\int\limits_0^\infty \mathrm{d}t\,\mathrm{e}^{-\sigma^2 t}\,t^{s-2} \,\theta_4\left(\frac{1}{2}\mathrm{i}\beta\mu, \mathrm{e}^{-\frac{\beta^2}{...
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1answer
35 views

Determine values of $\theta$ for which $\arg(z-4+2i)=\theta$ and $|z+6+6i|=4$ have no common solutions

So there is this question that's asking for a "range of values for theta from $-\pi$ to $\pi$, for which $\arg(z-4+2i)=\theta$ and $|z+6+6i|=4$ have no common solutions." I'm not really sure ...
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1answer
471 views

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}^{\...
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1answer
90 views

How is the integral form of Ramanujan theta function derived?

Ramanujan theta function defined as-$$f(a,b)=\sum_{n=0}^\infty a^{\frac{n(n+1)}{2}}b^{\frac{n(n-1)}{2}}$$ And it's integral representation:$$f(a,b)=1+\int_0^\infty \frac{2ae^{-t^{2}/2}}{\sqrt{2\pi}}\...
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0answers
64 views

The estimate for $\sum_{0}^{\infty} e^{-kn^2}-e^{-k(n+1/2)^2}$

I'm trying to give estimate on the following infinite summation: $\sum_{n=-\infty}^{\infty} (e^{-kn^2}-e^{-k(n+\frac{1}{2})^2})$, and k is some fixed positive number. The first term in the summation ...
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0answers
47 views

A closed form for these non-elementary, generalized relatives of the geometric series

I usually provide more details to the questions I ask on this site, but for this specific question I can't even wrap my head around how to even attempt solving it. By the way, this question has no ...
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0answers
39 views

$\sum_i^{\infty}e^{(-(2i+1)^2a)}(2i+1)^2$ where a is is a positive real number

I have asked this question two days ago. One helpful person in the comments directed me to the Jacobi Theta functions and I was able to solve the rest myself. Starting from that point, I have: $\sum_{...
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0answers
36 views

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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1answer
71 views

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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1answer
84 views

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 ...
2
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1answer
141 views

how to prove these formulas about infinite product?

Recently , I read one paper titled 'Modular equations and approximations to π' by Ramanujan, in which there are some formulas for $q=\pi i \tau$( where $\tau=x+yi, y>0$, hence $|q|<1)$ : $$\...
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1answer
71 views

Theta functions identites

I have been reading Rick Miranda's Book on Riemann surfaces and to indroduce meromorphic functions on the complex torues $\mathbb{C}$\ $L$ he talks about theta functions. I was able to see that $\...
2
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1answer
277 views

Zeros of the Jacobi Theta function

How do you obtain all the zeros in $z$ of the Jacobi Theta function $$\vartheta(z) = \sum_{n} e^{\pi i n^2 \tau + 2\pi i n z} \, ?$$ Probably the easiest way is to just read them of the Jacobi-Triple ...
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57 views

The Four Square Theorem and Integral Apollonian Circle Packings, is there any connection?

I have been studying theta-functions and made an interesting observation which I have a question about QUESTION: Is there a more intuitive, in particular a mostly geometric way, to prove the four ...
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1answer
39 views

Transformation formula for Theta-series

I am currently reading Weil's book : "Elliptic Functions According to Eisenstein and Kronecker" and in page 56 he uses the well-known transformation formula for theta series $$\sum\limits_{\mu} ...
3
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1answer
328 views

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 ...
2
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1answer
104 views

Is $\sum_{n \in \mathbb{Z}} e^{-(n-\mu)^2/2\sigma^2} \le \sum_{n \in \mathbb{Z}} e^{-n^2/2\sigma^2}$ for all $\mu$ and all $\sigma$?

I have been looking at discrete Gaussian distributions and arrived at the following conjecture. I would greatly appreciate a proof (or disproof). Conjecture. Let $\mu \in [0,1]$ and $\sigma^2 > ...
2
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1answer
109 views

Weierstrass elliptic function identity

For a lattice $\Lambda = [\lambda_1, \lambda_2] \subset \mathbb C$, the Weierstrass $\wp$-function defined as \begin{equation} \wp(z) = \frac{1}{z^2} + \sum_{\lambda \in \Lambda \setminus \{0\}} \...

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