Questions tagged [theta-functions]
For questions about $\theta$ functions (special functions of several complex variables).
257
questions
3
votes
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Remark 1: On p.384 of volume 3 of Gauss's Werke, which is a part of an unpublished treatise on the arithmetic geometric mean, Gauss makes the following remark: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 ...
1
vote
0answers
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 ...
0
votes
0answers
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 ...
6
votes
1answer
94 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 ...
0
votes
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)= $$\sum_{i=1}^ni^2=1^2+2^2+3^2+…+n^2=\frac{n(n+1)(2n+1)}{6}$$
...
0
votes
0answers
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
$$\vartheta(z;q=e^{\pi i \...
1
vote
0answers
31 views
Jacobi Quintuple Product Identity
I have to show the Jacobi Quintuple Product Identity
$$\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^{\...
2
votes
0answers
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 ...
1
vote
0answers
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:
$$\vartheta'_1 (0) = \vartheta_2 (0) \...
5
votes
0answers
159 views
Validity of $\ln z=\frac{\pi}{\operatorname{AGP}(\theta_2^2(1/z),\theta_3^2(1/z))}$
Definitions
For the definitions of $\operatorname{AGP}$ and $\operatorname{AGO}$, see here or here. $\theta_2(z)$ and $\theta_3(z)$ are defined as follows:
$$\theta_2(z)=\sum_{n=-\infty}^\infty z^{(n+...
4
votes
0answers
28 views
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 ...
3
votes
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....
0
votes
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 ...
0
votes
0answers
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
$$\sum _{n=-\infty }^{\infty } e^{-n^2 \pi x} = \frac{1}{...
0
votes
0answers
49 views
Square of Jacobi theta function is sum of hyperbolic secant?
I'm presently reading Henri Cohen's Introduction to Modular Forms (https://arxiv.org/pdf/1809.10907.pdf) and I'm trying to do exercise 1.5, which partially entails showing that:
$T_2(a)\equiv\sum_{n=-\...
1
vote
0answers
51 views
Symbolic calculation vs. desmos
Context:
I am working on getting a recurrence of the form
$$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=\...
3
votes
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 $...
3
votes
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 ...
3
votes
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 ...
4
votes
0answers
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
votes
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 ...
0
votes
0answers
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 ...
3
votes
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 ...
1
vote
0answers
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 $$ \sum_{n\in\mathbb{Z}}\mathrm{e}^{\mathrm{i}n\theta}\left(...
4
votes
1answer
107 views
Logarithmic differentiation of a complicated product
In this answer, it says that one may take the logarithmic derivative w.r.t. $z$ on both sides of the equation
$$(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)_\...
2
votes
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
$$
\int_0^{\infty} e^{-t^2/2} \left[
\frac{e^{2\pi} - \cos\left(\sqrt{2\pi} t
\right)}{e^{4\pi} - 2 e^{2\pi} \cos\left(\sqrt{2\pi} t\right) + 1}
\right] dt = \...
1
vote
2answers
101 views
Integral with theta function
$$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|...
5
votes
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
\...
0
votes
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
votes
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 $$\...
12
votes
2answers
298 views
Summation of $\sum_{n=0}^{\infty}a^nq^{n^2}$
I am trying to find the result for the sum of the form
$\sum_{n=0}^{\infty}a^nq^{n^2}$.
The special case for $a=1$ is easily given by $\vartheta(0,q)$, where $\vartheta(z,q)$ is the third Jacobi Theta ...
1
vote
1answer
103 views
What are specific proofs of Jacobi Triple Product Identity?
I am looking for the Special Proofs.
Here is a reference from MSE.
Motivation for/history of Jacobi's triple product identity
I also know that a simple proof via Functional Equation from the book ...
0
votes
0answers
43 views
Combined Integral transformations (Mellin + Laplace)
I'm looking for the solution of the following integral $$\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}{...
1
vote
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 ...
20
votes
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 $$\frac{1}{2}+\sum_{n=1}^{\...
1
vote
1answer
90 views
How is the integral form of Ramanujan theta function derived?
Ramanujan theta function defined as-$$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}}\...
1
vote
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 ...
2
votes
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 ...
0
votes
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_{...
1
vote
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 ...
0
votes
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 ...
-1
votes
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
votes
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 $|q|<1)$ :
$$\...
0
votes
1answer
71 views
Theta functions identites
I have been reading Rick Miranda's Book on Riemann surfaces and to indroduce meromorphic functions on the complex torues $\mathbb{C}$\ $L$ he talks about theta functions. I was able to see that $\...
2
votes
1answer
277 views
Zeros of the Jacobi Theta function
How do you obtain all the zeros in $z$ of the Jacobi Theta function
$$\vartheta(z) = \sum_{n} e^{\pi i n^2 \tau + 2\pi i n z} \, ?$$
Probably the easiest way is to just read them of the Jacobi-Triple ...
4
votes
0answers
57 views
The Four Square Theorem and Integral Apollonian Circle Packings, is there any connection?
I have been studying theta-functions and made an interesting observation which I have a question about
QUESTION: Is there a more intuitive, in particular a mostly geometric way, to prove the four ...
0
votes
1answer
39 views
Transformation formula for Theta-series
I am currently reading Weil's book : "Elliptic Functions According to Eisenstein and Kronecker" and in page 56 he uses
the well-known transformation formula for theta series $$\sum\limits_{\mu} ...
3
votes
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
votes
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
votes
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\}} \...
