Questions tagged [theta-functions]
For questions about $\theta$ functions (special functions of several complex variables).
138
questions with no upvoted or accepted answers
23
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Convexity of $\theta(q)$
Define Jacobi's (fourth) theta function with argument zero and nome $q$:
$$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$
plot of the function via Wolfram|Alpha
plot of the function via Sage
I ...
11
votes
0
answers
699
views
Global sections of vector bundles on a complex elliptic curve and analytic functions
Let me fix an elliptic curve $E$ over complex numbers with distinguished point $x \in E$. Thanks to Atiyah we know everything about discreet parameters of vector bundles and its moduli spaces. But I ...
9
votes
1
answer
473
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How to estimate a specific infinite sum
Let $M$ be an $n$ by $n$ matrix with each diagonal element equal to $k$ and each non-diagonal element equal to $k-1$ where $n$ and $k$ are positive integers. Let $k < n$ and we can assume both $k$ ...
9
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0
answers
1k
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How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$
I'd like to simplify
$$\newcommand{\bigk}{\mathop{\huge\vcenter{\hbox{K}}}}B(s)=\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_{k}(s)}{f_{k}(s)}\right)^{-1}$$ to something of the form $$...
8
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0
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133
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A relation concerning the "sum of squares" counting function $r_2(n)$
Let $r_2(n)$ denote the number of ways in which a positive integer $n$ can be expressed as the sum of squares of two integers. Here the sign as well as order of summands matters. Also by convention we ...
8
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0
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657
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Why do the Jacobi theta functions have a natural boundary?
The Jacobi theta functions, like
$$
\theta_3(z,q)=1+2\sum_{n=0}^\infty q^{n^2}\!\cos(2nz)
,
$$
look relatively innocent in how they handle the 'nome' $q$, a complex parameter that shapes the ...
7
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128
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Primes of the form $x^2 + n y^2$ using theta function
It is well known that if $p$ is a prime, then $p$ can be written in the form $x^2 + n y^2$ under certain congruences conditions. For example,
\begin{align}
p = x^2+y^2 &\Leftrightarrow p = 2 \text{...
7
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0
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163
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On Jacobi theta functions and some curious evaluations
(This post has been modified to include $\vartheta_2$.) The Jacobi theta functions are well-known for the identity,
$$\vartheta_2^4 - \vartheta_3^4 +\vartheta_4^4 = 0 $$
so this post will involve 4th ...
6
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0
answers
413
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Jacobi Theta Function on the Unit Circle - Is there a Limit in the Distribution Sense?
The third Jacobi theta function
$$\theta_{3}\left(z,q\right)=1+2\sum_{n=1}^{\infty}q^{n^{2}}\cos\left(2\pi n z\right)$$
appears in the study of path integrals in QM. Specifically in the problem of a ...
6
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0
answers
138
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Calculating $\Theta$ series of $E_8$ Lattice
I'm trying to calculate the $\Theta$ series of $E_8$ lattice, using the following Gram matrix (the Cartan Matrix of $E_8$):
$$\left(\begin{matrix}
2 & 0 & 0 & -1 & 0 & 0 & 0 &...
6
votes
0
answers
379
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Evaluating the alternating hyperbolic series $\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k \sinh (\pi k)}$
You can evaluate the alternating hyperbolic series $$\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k^{2n-1} \sinh (\pi k)}$$ for any positive even value of $n$ by integrating the function $$\frac{\pi \csc (\pi ...
6
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2
answers
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Jacobi Theta Functions?
For the Jacobi theta function $\vartheta_3(z|\tau)$ there exists an equality (by Whittaker & Watson)
\begin{equation}
\vartheta_3(z|\tau) = \sum_{n=-\infty}^{\infty} e^{n^2 \pi i \tau + 2 n i z} =...
5
votes
0
answers
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Is this function an alternative solution to the nonlinear pendulum?
Is this function an alternative solution to the nonlinear pendulum?
Introduction
I am working with the differential equation of the frictionless nonlinear pendulum:
$$\ddot{\theta}(t) + b\,\sin(\theta(...
5
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0
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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+...
5
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0
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244
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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
\...
5
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0
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106
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Has the idea of a 'hyperbolic Theta function' been studied?
Consider a uniform $\{m,n\}$ tiling of the hyperbolic plane, for convenience with one vertex at the origin (and also for convenience, normalize the edges to have unit hyperbolic length). Then there ...
4
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0
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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 ...
4
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0
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96
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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 ...
4
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0
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303
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Relationship between $\theta$ functions and number fields.
I'm trying to have a clear picture of the relationship of theta functions and $L$-functions, and the geometric objects they relate to.
Firstly, I know that $\theta$-functions arise as sections of ...
4
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0
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276
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Closed form of $\sum_{n=1}^\infty q^{- n^2} z^n$
In this question the summation goes from $-\infty$ to $\infty$ and the answer has a pretty "good" closed form.
Now I came across the sum $\sum_{n=1}^\infty q^{-n^2} z^n$ where $|z|<1$ and I don't ...
4
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0
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136
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How many integer solutions are there to $x^4+y^4-x^2y^2=n$. Is there a generating function for this?
It would be kind of cool to get a closed form for the number of integer solutions $$x^4+y^4-x^2y^2=n$$ which we will let $\phi_n$ denote.
It would be cool because we could exploit $\sum_{n=1}^N\...
4
votes
0
answers
149
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Evaluate $ \frac{1}{(q)_\infty} \sum_{m \in \mathbb{Z}} q^{\frac{m^2}{2}} (-q^{-\frac{1}{2}}x)^m y^m(q^{1-m}y^{-1};q)_\infty $
This identity is taken from a physics paper [1] stated without proof, on page 43.
$$ \frac{1}{(q)_\infty} \sum_{m \in \mathbb{Z}} q^{\frac{m^2}{2}} (-q^{-\frac{1}{2}}x)^m y^m(q^{1-m}y^{-1};q)_\infty
=...
4
votes
0
answers
98
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Reciprocal sums
How can I prove the following?
1)$$\sum_{k=0}^{\infty}\frac{1}{F_{2k+1}}=\frac{\sqrt{5}}{4}\theta_{2}^{2}(0,\frac{3-\sqrt{5}}{2}),$$
2)$$\sum_{k=0}^{\infty}\frac{1}{F_{k}^{2}}=\frac{5}{24}(\theta_{2}^{...
4
votes
1
answer
268
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About the sums $\sum_{n=1}^\infty x^{n^2}$ and $\sum_{n=1}^\infty \frac{x^n}{1+x^{2n}}$
Despite all my efforts trying to crack these, i haven't been able to do so. An approach that i've tried gives me somewhat of an asymptotic approximation, but still fails to produce the values near x=0....
4
votes
0
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88
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Is $\sum_n\exp(ian+ibn^2+icn^3)$ known in terms of anything else?
For arbitrary $a,b,c$, does the series
$$F(a,b,c)=\sum_{n=-\infty}^\infty\exp\left(ian+ibn^2+icn^3\right),$$
i.e. an evenly-weighed series of exponentials of cubic polynomials, converge to anything ...
4
votes
1
answer
115
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Injectivity of theta function
Let $\vartheta_{00}$ and $\vartheta_{01}$ be Jacobian theta functions (notations like on wikipedia). $$F:=\left\{ \tau \in \mathbb{C}: \mathrm{Im}(\tau)>0, \left| \mathrm{Re}(\tau)\right|<1, \...
4
votes
0
answers
442
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Modular forms on the theta group
The theta group $\Gamma$ is the subgroup of $SL(2;\mathbb{Z})$ generated by $T=\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ and $S=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.$ It ...
3
votes
0
answers
136
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Integral $\int_0^1\ln^2(t)\psi^8(-t)dt=\frac{7}{8}\zeta(3)$
I was working on the following Integral:
$$I=\int_0^1\ln^2(t)\psi^8(-t)dt=\frac{7}{8}\zeta(3)$$
Where,
$$\psi(q)=\sum_{n=-\infty}^{\infty}q^{n(2n+1)}$$
is a Ramanujan Theta Function.
Now it is well ...
3
votes
0
answers
129
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How did people come up with the product formula for Jacobi's theta function
I understand that the function $\Theta(z|\tau)$, with its definition
$$\Theta(z | \tau) := \displaystyle\sum_{n = -\infty}^{\infty} e^{\pi in^2\tau} e^{2\pi inz},$$
satisfies the product formula
$$\...
3
votes
0
answers
29
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$(\mathfrak{g}, K)$-module for covering group of $\mathrm{GL}_2(\mathbb{R})$
Harich-Chandra introduced the notion of $(\mathfrak{g}, K)$-module to study representations of a real Lie group $G$ via its Lie algebra $\mathfrak{g}$ and a maximal compact subgroup $K$.
One also has ...
3
votes
0
answers
99
views
Line bundle of complex tori
Let $\Lambda \subset V=\mathbb{C}^g$ be a lattice and $T_g=V/\Lambda$ be a $g$ dimensional complex torus.
According to the Appell-Humbert theorem, any line bundle $L$ of $T_g$ is isomorphic to the ...
3
votes
0
answers
74
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A "surprising" asymptotic inverse of $\vartheta _3(0,x)$
After this question of mine related to the problem of approximate solutions of
$$\large\color{red}{\operatorname{\vartheta}_3}\left(0,x\right)=k$$ when $k$ is large, continuing the previous work (just ...
3
votes
0
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80
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From root and weight lattices of SU(N) to $\theta$-functions as sections of a line bundle and $CP$-space
I have troubles to digest the following messages/discussions in the following work in p.10-12;
Which construct a map from the moduli space of flat connections $M_{\rm flat}=\mathbb{E} / {\mathfrak S}...
3
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0
answers
164
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Discrete theta functions and periodicity
I'm doing quantum mechanics and I have an eigenfunction which is a theta function. I then discretised it, since I want see if I can find the eigenvalues for the discrete case by finding the ...
3
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0
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566
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Definite integral of Jacobi Theta Function
I need to evaluate an integral that involves the Jacobi Theta Function $\vartheta_3(z,q)$ defined by MathWorld as
$$\vartheta_3(z,q)=\sum_{n\in\mathbb{Z}}q^{n^2}e^{2inz}.$$
Specifically, I wish to ...
3
votes
0
answers
839
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Implementation of Jacobi theta functions in Matlab
I need Jacobi theta functions for my Matlab program. The functions are not included in the predefined Matlab functions. Doing a simple Google search, I found a package developed by Moiseev I. in 2008. ...
3
votes
0
answers
124
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Degree of the Divisor of a Theta Function
Let $(\gamma_1, \gamma_2)$ be a base for a lattice $\Gamma$ in $\mathbb C$, and $\theta$ a theta function, ie an holomorfic function such that $\theta(z+ \gamma) = \theta(z)e^{2i\pi(a_\gamma z + b_{\...
3
votes
0
answers
98
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What is known about $\sum_{n=0}^{\infty} x^{n^3} $.
$f(x) =\sum_{n=0}^{\infty} x^{n^2}$ and similar "theta-type" functions are extensively studied.
They have many properties and occur in number theory , algebra (in particular solving the quintic ...
3
votes
0
answers
443
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Values of derivatives of Jacobi theta function
The Jacobi theta function is defined as:
$$\theta(x)=\sum_{n=-\infty}^{\infty}\exp(-\pi n^2 x)\text{ }, x>0$$
On wikipedia.org, we can find close-form expressions for the values of $\theta(k)$,$...
3
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0
answers
285
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A Dedekind eta function sum of form $y_0^k+y_1^k+y_2^k+y_3^k+... = 0$
Given the Dedekind eta function $\eta(\tau)$. Define,
$y_p = e^{\pi i p/6}\,\eta(\tfrac{\tau+2p}{5})$
Prove the multi-grade identity [1],
$y_0^k + y_1^k + y_2^k + y_3^k + y_4^k + (\sqrt{5}\,\eta(5\...
2
votes
0
answers
36
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How does Jacobi form lives projectively on the torus $\mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})$?
I saw the statement in the question from the book Moonshine Beyond the Monster.
We are given the definition : a group hom $\rho : G \rightarrow PGL(V)$ is called a projective representation.
I can't ...
2
votes
0
answers
32
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Elliptic functions by Eisenstein-Kronecker
In $\textit{Elliptic Fuctions according to Eisenstein and Kronecker}$, chap VIII, section 13 by A.Weil there is the following problem
For any integer $k \geq 0$ and $z, w \in \mathbb{C}$, the function ...
2
votes
0
answers
79
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Functional equation for $\theta$ function using functional equation of $\zeta(s)$
Let $$\theta(t) = \sum_{n \in \mathbb Z}e^{-n^2 \pi t}.$$ We can derive a functional equation for $\theta$ using Poisson summation formula: $$\theta(1/t) = \sqrt t \theta(t). $$
Riemann uses the above ...
2
votes
0
answers
59
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The theta function on Weierstrass normal form
We want to write down the theta function of the point on Weierstrass normal form.
Now, we can embed an elliptic curve to projective plane by$E=\mathbb{C}/(\tau\mathbb{Z}+\mathbb{Z})\to \mathbb{P}^2$
...
2
votes
0
answers
78
views
A sum related to the theta function
I am interested in the serie
$$\sum_{n=1}^\infty(2n+1)e^{-tn(n+1)}$$
for something I am working on. Wolfram alpha gets stuck on this problem, but it gives the following result for a closely related ...
2
votes
0
answers
84
views
Questions about convergence related to Theta functions.
Let $H_n$ be the set of matrices $z\in \mathbb{C}^{n\times n}$ which are symmetric with a positiv-definite imaginary part.
Let's assume that it is already known and proved that
$\vartheta_a(z)=\sum_{g\...
2
votes
0
answers
66
views
Adelic theta function over function fields
I saw the following claim on Godement-Jacquet's classical book "Zeta functions of simple algebras": (on page 153) Let $F$ be a global function field, $\Phi$ a Schwartz function on $\mathbb{A}...
2
votes
0
answers
51
views
Need help understanding proof of the functional equation for the theta function.
I am following the proof of theorem 1.3 above until at the point the author finds the recurrence relationship $a_n = a_{n-k}e^{b - 2\pi ni\tau}$.
When $k = 0$ we have $a_n = a_{n}e^{b - 2\pi ni\tau} \...
2
votes
0
answers
109
views
Is there any definition for this series function $f(s)=\sum_{n=1}^\infty e^{-n^s}$?
What I am asking for is if there is any theory related to this real series $f(s)=\sum_{n=1}^\infty e^{-n^s}$ and $s\ge 1$.
As far as I know, if $s = 1$, it's a simple geometric series, and if $s = 2$, ...
2
votes
0
answers
57
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Weierstrass factorization of $\theta_1(z|\tau)$ in $z$
Suppose $f$ is an entire function and $f(0)\ne 0$. Let $z_1, z_2,\ldots$ be the zeros of $f$ and $p_1, p_2,\ldots$ be any sequence of nonnegative integers such that
$$r\gt 0\implies \sum_{n=1}^\infty \...