Questions tagged [theta-functions]
For questions about $\theta$ functions (special functions of several complex variables).
308
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A sum related to the theta function
I am interested in the serie
$$\sum_{n=1}^\infty(2n+1)e^{-tn(n+1)}$$
for something I am working on. Wolfram alpha gets stuck on this problem, but it gives the following result for a closely related ...
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Translations of elliptic theta functions
From Gradshteyn and Ryzhik's Table of Integrals, Series, and Products textbook I have the following half-period translation identities
$$H(K(k)+i\alpha)=H_1(i\alpha)$$
$$\Theta(K(k)+i\alpha)=\Theta_1(...
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4
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A theta function identity involving $\vartheta_2(q^3),\vartheta_3(q^3)$
How can we verify the theta function identity?
$$
\left ( \vartheta_2(q)^2+3\vartheta_2(q^3) ^2\right )\left ( \vartheta_3(q)^2+3\vartheta_3(q^3)^2 \right )=4\vartheta_2(q)^2\vartheta_3(q)^2.
$$
Where ...
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Showing an equality for the KS-statistic with Jacobi theta functions
I recently read about the Kolmogorov-Smirnov statistic and on wikipedia it is stated that the cdf of the random variable $K$ is given by
$$
Pr(K\leq x) = 1-2\sum_{n=1}^{\infty} (-1)^{n-1}e^{-2n^2x^2} =...
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What is the exact value of $\sum\limits_{n=1}^\infty \frac{1}{F_n}$? [duplicate]
What is the exact value of $\sum\limits_{n=1}^\infty \frac{1}{F_n}$? $F_n$ denotes the $n^{th}$ Fibonacci number.
Wolframalpha gave me this answer: $$\sum_{n=1}^{\infty}\frac{1}{F_n}\ =\frac{1}{4}\...
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Do closed form expressions exist for $\int_0^1 \theta_3(x)\,dx$?
The Jacobi theta (or “thetanull”) function $\theta_3$ is defined by:
$$\theta_3(x)= \sum_{n \in \mathbb{Z}} \mathrm{e}^{-\pi n^2 x} = 1+ 2\sum_{n \in \mathbb{N}} \mathrm{e}^{-\pi n^2 x} \qquad \Re(x) &...
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Checking the parity of an infinite sum
While going through a paper I stumbled on the following equality :
$\begin{equation} \rho(\alpha)=\frac{1}{\pi}\sum\limits_{k\in\mathbb{Z}}\frac{e^{2ik\alpha}}{\cosh(k\zeta)}=\frac{1}{\pi}\frac{\...
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Prove a relation between Theta functions and their derivatives to prove $\sum_{n=-\infty}^\infty \frac{1}{\cosh^2(\pi n )}$
In an attempt of proving
\begin{align*}
\sum_{n=-\infty}^\infty \frac{1}{\cosh^2(\pi n )}=\frac{1}{\pi}+\frac{\Gamma^4\left( \frac14\right)}{8 \pi^3} \tag{1}
\end{align*}
I found the following ...
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Primes of the form $x^2 + n y^2$ using theta function
It is well known that if $p$ is a prime, then $p$ can be written in the form $x^2 + n y^2$ under certain congruences conditions. For example,
\begin{align}
p = x^2+y^2 &\Leftrightarrow p = 2 \text{...
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On Jacobi theta functions and some curious evaluations
(This post has been modified to include $\vartheta_2$.) The Jacobi theta functions are well-known for the identity,
$$\vartheta_2^4 - \vartheta_3^4 +\vartheta_4^4 = 0 $$
so this post will involve 4th ...
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Why does the theta function appear in the study of complex tori?
Let $L$ be a lattice and $X = \mathbb{C}/L$ a complex tori.
To obtain a meromorphic function $f$ on $X$, it is standard to consider ratios of products of (translated) theta-functions, and in fact, any ...
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$(\mathfrak{g}, K)$-module for covering group of $\mathrm{GL}_2(\mathbb{R})$
Harich-Chandra introduced the notion of $(\mathfrak{g}, K)$-module to study representations of a real Lie group $G$ via its Lie algebra $\mathfrak{g}$ and a maximal compact subgroup $K$.
One also has ...
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Questions about convergence related to Theta functions.
Let $H_n$ be the set of matrices $z\in \mathbb{C}^{n\times n}$ which are symmetric with a positiv-definite imaginary part.
Let's assume that it is already known and proved that
$\vartheta_a(z)=\sum_{g\...
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Mellin transform of theta function $\theta$ to show $\zeta(-1)=-\frac{1}{12}$
Edit: I realize a lot of attention is pointed at the proclamation $\zeta(-1)=-\frac{1}{12}$. Im more interested in the derivation of the Zeta function using the methods describe below, thank you!
...
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Adelic theta function over function fields
I saw the following claim on Godement-Jacquet's classical book "Zeta functions of simple algebras": (on page 153) Let $F$ be a global function field, $\Phi$ a Schwartz function on $\mathbb{A}...
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Line bundle of complex tori
Let $\Lambda \subset V=\mathbb{C}^g$ be a lattice and $T_g=V/\Lambda$ be a $g$ dimensional complex torus.
According to the Appell-Humbert theorem, any line bundle $L$ of $T_g$ is isomorphic to the ...
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How to prove Gauss's identities on the action of the operator $f' = x\frac{df}{dx}$ on Jacobi theta functions?
P.444-445 of volume 3 of Gauss's Nachlass, in which Gauss wrote down an identity for the infinite series of $\vartheta^4_3(x)$ (this identity is the essence of Jacobi's four squares theorem), include ...
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How to prove an identity of Gauss for the logarithm of Jacobi theta function?
P.444 of volume 3 of Gauss's Nachlass includes the following identity for the logarithm of $\vartheta_3(x)$:
$$\mathbb{log}(1+2x+2x^4+2x^9+\cdots) = \frac{2x}{1+x}+\frac{2x^3}{3(1+x^3)}+\frac{2x^5}{5(...
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Expressing inverse beta regularized in terms of Riemann/Siegel theta
I am new to the Riemann/Siegel Theta function, but it represents many special cases of Inverse Beta Regularized $\text I^{-1}_s(a,b)$. The Riemann theta function can represent any Abelian function, ...
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Numerical convergence of a sum
I want to study the convergence of the function given below.
$$ \sum_{n = 1}^{\infty} e^{-\Gamma^2 n^2} $$
The $n$ are integers!
Here, if we check numerically, the function converges based on $\Gamma^...
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2
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How to calculate zeta-like function $\sum_{n=k}^\infty \frac{1}{(n+a)^s}$
Here I want to calculate a zeta-like function $\sum_{n=k}^\infty \frac{1}{(n+a)^s}$ where $k \in \mathbb{N}$, $s>1$ and $0<a<1$. I usually calculate the Riemann zeta function by the Poisson ...
- 79
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Intuition behind theta function L-series correspondence
I know that we can analytically continue $L$-series by taking the Mellin transform of a sutiable theta function. Technically we need a transformation law for the theta function at $0$ and $\infty$ as ...
user427380
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2
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A series in terms of the Jacobi theta fuctions
On Wikipedia, one can find an expression of the series
$$
\sum_{n=1}^\infty \frac{n^pq^n}{1-q^n}
$$
in terms of the Jacobi theta functions for $p=3,5,7$. I'm looking for an expression of this series ...
- 1,589
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Summation in the form of Jacobi Theta Function
TL;DR: My summation should give the same result when it is expressed as the Jacobi Theta function. It gives the same results for some set of inputs but then gives exactly the half of it for other set ...
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Investigating limit of theta like series
Let $v>0$. I want to prove
$$\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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0
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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 $\theta(z) = \sum_{n \in \mathbb{Z}}e^{2\pi in^{2}z}$:
$$\theta(\gamma z) = \left(\frac{c}{d}...
user427380
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answers
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Need help understanding proof of the functional equation for the theta function.
I am following the proof of theorem 1.3 above until at the point the author finds the recurrence relationship $a_n = a_{n-k}e^{b - 2\pi ni\tau}$.
When $k = 0$ we have $a_n = a_{n}e^{b - 2\pi ni\tau} \...
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votes
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A convergence lemma for adelic zeta function in automorphic forms
I'm reading Godement-Jacquet's classic Zeta functions of simple algebras (1972, Springer). On page 153, the first line: " We also $\textbf{take for granted}$ the following lemma (numbered 11.3)......
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Integral of a sin function over the momentum space resulting in $\delta$ function
I am reading a paper by Faddeev and Kulish, whose name is asymptotic conditions and infrared divergences in QED (the paper is not important for the question I think, but I will include the link: https:...
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In which dimensions do there exist inequivalent lattices with the same theta function?
Equivalently, "Is a lattice determined by the distances (with multiplicities) of its points from the origin?"
By a lattice $L$ I mean a discrete additive subgroup of Euclidean space $\mathbb{...
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Verifying Modularity for the Theta Function
I was trying to verify modularity of the theta function
$$\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) = \...
user427380
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0
answers
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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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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:
$$\ddot{\theta}(t) + b\,\sin(\theta(...
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Gaussian integral using Euler/Jacobi theta function and $r_2(k)$ (number of representations as sum of 2 squares)
The Euler/Jacobi theta function (using the notation of this question) is $\vartheta_3(\tau) := \sum_{n\in \mathbb Z} q^{n^2}$ where $q = e^{2\pi i\tau}$ is the nome. The square $(\vartheta_3(\tau))^2$ ...
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Relation Between Jacobi's Theta Function and Weierstrass $\wp$ Function
I am reading Elliptic Curves by Moll and McKean and it defines Jacobi's theta function on the lattice $\Gamma = \{n+m\omega \mid m,n \in \mathbb{Z}\}$ for a $\omega \in \mathbb{H}$ as below:
$$\...
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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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1
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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
$$ \prod_{k=1}^{\infty} \left[ 1+\frac{-1+i\sqrt{3}}{2}x^k + \frac{-1-i\sqrt{3}}{2}x^{2k}\right] = \...
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A "surprising" asymptotic inverse of $\vartheta _3(0,x)$
After this question of mine related to the problem of approximate solutions of
$$\large\color{red}{\operatorname{\vartheta}_3}\left(0,x\right)=k$$ when $k$ is large, continuing the previous work (just ...
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Is there an analytic expression for $\sum\limits_{n=0}^{\infty}x^{n^2}$
In statistical mechanics I often come across average energies of the form:
$$\begin{equation}
\langle\epsilon_n\rangle=\alpha \sum_{n=0}^{\infty}n^2e^{-\alpha n^2}
\end{equation}$$
where $\alpha$ is ...
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0
answers
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Connection between elliptic integrals and theta functions
I have read about connections between elliptic integrals and their connections to the Jacobi theta functions, like $\theta_3^2(q) = \frac{2}{\pi}K(k)$, where $q=e^{-\pi\frac{K’(k)}{K(k)}}$, but how ...
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Does this Auxiliary Fresnel Sum=$\frac1{2\sqrt2\pi}\int \limits_0^\infty \frac{\vartheta_3\left(e^{-\frac{\pi x}2}\right)\sqrt x}{x^2+1}dx +\frac14 $?
$$\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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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
$$\color{red} {\operatorname{\vartheta}_3\left(0,x\right)=k} \qquad \...
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About the asymptotic behavior of specific Jacobi $\theta$ function $\operatorname{\vartheta}_3\left(0;x\right)$ when $x\to{1-}$.
Since $\displaystyle\sum_{n=1}^\infty{x^{n^2}}=\dfrac{\operatorname{\vartheta}_3\left(0,x\right)-1}2$ for $x\in\left(0,1\right)$ (just in case), it suffices to consider the former below. (Another ...
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Closed form for the indefinite integral of the Jacobi Theta function
I am interested in the indefinite integral of $\vartheta_3(q;0)$. A lazy result gives us
$$
\int\vartheta_3(q;0)\mathrm{d}q=q+2\sum_{n=1}^\infty \frac{q^{n^2+1}}{n^2+1}+C.
$$
While there is nothing ...
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0
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Proof of Jacobi fraction expansion in Triple Product Proof
In the proof of Jacobi's triple product identity by Jacobi, he considers the infinite product; $\frac{1}{(1-qz)(1-q^2z)...}$
and expands it into 1 + $\frac{B_1z}{(1-qz)}$ + $\frac{B_2z^2}{(1-qz)(1-q^...
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1
answer
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How to choose witnesses for asymptotic growth?
I'm struggling with how to choose witnesses for asymptotic growth. Specifically, here is the problem I'm working on and what I have done so far:
Prove: $$4n^5 – 50n^2 + 10n \in \Theta(n^5)$$
$$0 \leq ...
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0
answers
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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: $$f(x,y)=\sum_{n=1}^\infty \sum_{k=1}^\infty\exp\big(-n^xk^y\...
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answer
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What is the asymptotics of: $\Re\left(\frac{\zeta \left(1+\frac{1}{c}\right) \zeta (s+i t)}{\zeta \left(s+i t+\frac{1}{c}+1-1\right)}\right)$?
What is the asymptotics of the function $f(s,t,c)$:
$$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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