Questions tagged [theta-functions]
For questions about $\theta$ functions (special functions of several complex variables).
0
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Properties of the Theta function
Define the $\vartheta :\mathbb{R}^+ \to \mathbb{R}$ by
$$\vartheta (s) = \sum_{m=-\infty}^{\infty} e^{-\pi m^2s}$$
Is this a smooth ($C^{\infty}$) function? I would like to think so, but I'm not ...
5
votes
1answer
72 views
Addition formula for elliptic integral of second kind
Let $k\in(0,1)$ and the incomplete elliptic integral integral $E(u, k) $ be defined by $$E(u, k) =\int_{0}^{u}\operatorname {dn} ^2(t,k)\,dt\tag{1}$$ where $\operatorname {dn} (u, k) $ represents one ...
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0answers
33 views
Asymptotic behavior of $2\sum_{i=0}^n 4^i$
I know that the summation
$$2\sum_{i=0}^n 4^i$$
exhibits $\theta(2^{n*ln(n)})$ asymptotic behavior. How do I prove this, however?
I know that as $n\rightarrow \infty$ the value become ~ $2*(n+1)*4^...
1
vote
1answer
69 views
Characteristic functional equation of a Theta Function
Define the following as a "simple" theta function
$$ \vartheta(q) = \sum_{n=0}^{\infty} q^{n^2} = 1 + q + q^4+q^9+ \ ...$$
Defined on the open unit circle on the complex plane.
I'm trying to find ...
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0answers
17 views
How to prove that a division of $\vartheta(z:\tau)$ functions has no poles
How do I prove that $f(z) + f(iz)$ has no poles? where $$f(z)=\dfrac{\vartheta^2(0;i)\vartheta^2(z+ \frac{1}{2};i)}{\vartheta^2(\frac{1}{2};i)\vartheta^2(z;i)}$$ and $$\vartheta(z;\tau)=\sum\limits_{n=...
2
votes
1answer
38 views
Jacobi Triple Product using Euler's identities
I recently came across a simple proof for Jacobi's triple product (here), in the proof Andrews assumes two identities:
$$
E_1 = \prod_{n=0}^{\infty}(1+x^nz) = \sum_{n=0}^{\infty}\frac{x^{n(n-1)/2}z^n}...
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0answers
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Lattice associated to 4th Jacobi theta function?
For a lattice (specifically the dual lattice of a torus) there is associated a theta function
$ \theta_{\Gamma}(w)=\sum_{\gamma\in\Gamma}w^{||\gamma||^2},\text{ where $w=e^{-4\pi^2t}$ and $t\in(0,\...
2
votes
1answer
54 views
Theta function squared is a weight $1$ modular form
Let
$$\vartheta(\tau) = \sum_{n\in\mathbb{Z}}e^{\pi in^2\tau}.$$
I know that $\vartheta$ satisfies the transfromation properties
$$\vartheta(\tau + 2) = \vartheta(\tau), \quad \vartheta\left(-\frac{1}{...
2
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3answers
67 views
Asymptotics of Jacobi's third theta function.
For $z\in \mathbb{C}$ and $\tau \in i\mathbb{R}_{+}$ consider the function
$$ \theta_3(z;\tau)=\sum_{n\in\mathbb{Z}}\exp\left(2\pi i n z +\pi i \tau n^2\right) $$
This function satisfies the well ...
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0answers
20 views
What is a closed form partial sum formula for the q-digamma function?
It is known that the partial sum of the digamma function can be expressed in closed form, but what of the q-digamma function?
$$\sum_{x=1}^n\psi_q{\small(x)}\ =\ ?$$
$$q\in\mathbb N$$
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1answer
26 views
What does it mean when the exponent is beneath the base number?
I am familiar with the exponential function and the power function.
Exponential Function: $y=3^x$
Power of Function: $y=x^3$
But what do the numbers under the theta in this image represent?
$h_𝜃(...
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1answer
56 views
Infinite sum of 1/sin^2 and theta function
In studying some physical propagator, I came across the following sum
$$
\sum_{n = -\infty}^{+\infty} \frac{ a^n }{ \sin^2(z + n \pi \tau) }\ .
$$
Obviously, my question is how to evaluate this sum.
...
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0answers
52 views
Double sum of Lambert series: Partial sum in closed form desired!
We desire the things stated in the title for:
$\sum_{k=2}^m \sum_{n=1}^{k-1} {q^n\over {1-q^n}}$
Some things I've looked into that may be of some help:
The first sum is just a truncated (partially ...
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0answers
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Cutting out a Riemann surface inside its Jacobian variety
After choosing a base point $P_{0}$ in a compact Riemann surface $X$ of genus $g$, the Abel-Jacobi map gives an embedding of $X$ into its Jacobian variety $Jac(X).$ This map can also be extended to ...
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0answers
233 views
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 ...
3
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0answers
23 views
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}...
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0answers
20 views
Finding doubly periodic solutions to partial differential equations
Say I have a certain PDE in real variables $x$ and $y,$ which might be nonlinear, so that we can't necessarily just throw a Fourier series at it.
By way of some intuition, let's say, I have a very ...
1
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1answer
61 views
Transformation of a theta function
Given $\Theta(\tau)=\sum_{n \in \mathbb Z}exp(2\pi in² \tau)$ and $\tau \in \mathbb H$
I am trying to prove the following identity:
$\Theta(-\frac{1}{2\pi})=\sqrt{\frac{\tau}{i}}\Theta(\frac{\tau}{2}...
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0answers
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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 ...
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0answers
99 views
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$$ which we will let $\phi_n$ denote.
It would be cool because we could exploit $\sum_{n=1}^N\...
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0answers
49 views
Theta function equation
I have been trying to prove an equation of a theta function.
I understood that it some how related to the Poisson summation formula, but no luck. Any help would be appreciated.
the exercise
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0answers
32 views
Explicit Values of the Jacobi Theta Function [duplicate]
The sum $\sum_{n=-\infty}^{\infty}\exp(-\pi n^2) $, or $\vartheta(0;i)$ (Jacobi Theta Function) has a closed form solution of $\frac{\pi^{\frac{1}{4}}}{\Gamma(\frac{3}{4})}$ but nowhere have I been ...
2
votes
2answers
75 views
Evaluation of the limit $\lim_{q\rightarrow 1} \frac{\phi^5(q)_{\infty}}{\phi(q^5)_{\infty}}$
Given the Euler function $\phi(q)=\prod_{n = 1}^{\infty}(1-q^{n})$ which is a modular form where $q=\exp(2\pi i \tau)$, $|q|\lt1$
Then what is the limit $\lim_{q\rightarrow 1}\frac{\phi^5(q)_{\...
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0answers
33 views
how to derive the formula of Jacobi theta function which is below
What is the derivative of Jacobi theta function which is:
$$\Theta_3(z;\tau)= \sum_{n=-\infty}^\infty \exp(i\pi\tau n^2)\exp(2ni z)$$
and find its zeros.
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0answers
32 views
Analytic continuation of Riemann Theta function to positive semidefinite matrices
The Riemann $\Theta$-function is defined as
$$Θ(z|Ω)=\sum\limits_{q∈Z^N}e^{πiq⋅Ωq+2πiq⋅z},$$
where $\Omega$ has positive definite imaginary part to ensure convergence. In a particle physics ...
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votes
2answers
257 views
How are the Jacobi Theta Functions analogous to the Exponential Function?
On the Wolfram MathWorld page on Jacobi Theta Functions, it says that the Theta Functions are elliptic analogues of the exponential function. Is this because they satisfy certain properties that the ...
4
votes
1answer
165 views
What's the connection between $\theta$ series and the number of integer solutions on a curve? Proof Verification
I just started learning about theta series and am now flexing my muscles. I hope that everything looks good. This is a request for a proof verification. I am asking for what the proper name in the ...
3
votes
1answer
202 views
How many integer pairs satisfy the ellipse $x^2+ay^2=r?$
How many integer pairs satisfy the ellipse $x^2+ay^2=r?$
What I have discovered thus far:
This post is largely to document the thinking that I have already done... I know that this can be frowned ...
2
votes
0answers
67 views
Clarification of Landau's Mechanics, Chapter.6 & Whittaker, Analytical Dynamics Chapter.6 needed
I am currently working through Landau's mechanics book and I am struggling to get my head around a solution provided in Chapter.6 relating to the asymmetrical top.
I fully comprehend how Landau ...
3
votes
1answer
123 views
How many integer solutions are there on an $n$ dimensional hypersphere of radius $\sqrt{r}$ centered at the origin?
Let $\phi(n,r)$ be the number of integer solutions of
$\sum\limits_{i=1}^n x_i^2=r$.
Then $\phi(2,r)=4\sum\limits_{d|r}\chi(d)$ where $\chi (x)=sin(\frac{\pi x}{2})=\cases{ 1\text{ when }x\cong 1 \...
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2answers
64 views
$\sum\limits_{\mathbb{d|n}}{f(d)}=\sum\limits_{\mathbb{d|n}}{g(d)}\implies f(n)=g(n)?$
Question:
Is it true that if for functions $f,g$ which map naturals to naturals
For all natural numbers n, we have $f(n)=g(n) \iff$ for all natural
numbers n we have
$\sum\limits_{\mathbb{d|...
2
votes
1answer
70 views
Properties of $\frac {\theta_{1}''(z|\tau)}{\theta_{1}(z|\tau)}$?
I'm looking for references concerning the properties of the function $\frac {\theta_{1}''(z|\tau)}{\theta_{1}(z|\tau)}$ where $\theta_{1}(z|\tau)$ is a Jacobi theta function defined here. I am trying ...
1
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0answers
61 views
$\Theta$ function in terms of Weierstraß $\sigma$ function?
Let $\Theta$ function be the function associated to a lattice $\Lambda=\oplus_{i\leq 2}Z\lambda_i\subset C$ of $C$ with transformation property defined as $\lambda\in\Lambda, \Theta(z+\lambda)=\Theta(...
2
votes
1answer
55 views
$\theta(z) = \sum_{n=-\infty}^{+\infty} e^{\frac{-n^2}{2}}e^{inz}$ How to show $\theta '(\frac{i}{2}) = \frac{-i}{2} \theta (\frac{i}{2})$?
$$\theta(z) = \sum_{n=-\infty}^{+\infty} e^{\frac{-n^2}{2}}e^{inz}$$
How to show $\theta '(\frac{i}{2}) = \frac{i}{2} \theta (\frac{i}{2})$?
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0answers
50 views
Hecke Operators and Eigenfunctions, Fourier coefficients
The problem statement, all variables and given/known data
Consider the action of $T_2$ acting on $M_k(\Gamma_{0}(N)) $, and show that $\theta^4(n)+16F $ and $F(t)$ are both eigenfunctions.
Functions ...
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0answers
61 views
How to show that the theta function is smooth?
I was reading on the theta function
$$
\vartheta(x) = \sum_{n=-\infty}^\infty e^{-\pi n^2 x}\qquad x>0
$$
The author claims that $\vartheta$ is smooth. Is there an easy way to show that this is ...
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1answer
46 views
Basic $\theta$-function identity proof
For the $\theta$-function $$\theta (z) = \sum_{n \in \mathbb{z}} q^{n^2}e^{2\pi inz},$$
for $q$ given by $e^{\pi i\tau}$ for some $\tau \in \mathbb{C}$ with $Im(\tau) > 0$, suppose we've proved ...
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0answers
324 views
Asymptotic equivalent of $\sum_{n\ge0} q^{n^2}{x^n}$ as $x\to+\infty$
Let $q\in\Bbb C^*$ with $|q|<1$, define
$$f:x\mapsto\sum_{n\ge0} q^{n^2}{x^n}$$
I want to find an asymptotic equivalent of $f$ as $x\to+\infty$.
I found that
$$a\le|f(x)|\cdot\exp\left(\frac{\...
4
votes
1answer
72 views
The even-index reciprocal Lucas constant and $\sum_{n=1}^\infty \frac1{x_1^{2n}+x_2^{2n}}$
The sum of reciprocals of even index Lucas numbers has a nice closed-form in terms of theta functions,
$$\begin{aligned}S_e
&= \sum_{n=1}^\infty \frac1{L_{2n}}\\
&= \sum_{n=1}^\infty \frac1{\...
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vote
0answers
49 views
proving sigma = BigTheta (BigΘ)
I'm trying to solve a BigΘ problem, and could use a little guidance to make sure I'm on the right path.
So my question is to show that $\sum_{i=1}^{n} i^{15} = Θ(n^{16})$
I know that for something ...
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2answers
398 views
An interesting identity involving Jacobi $\theta_4$ and $\zeta(2)$
A recent question mentioned an integral identity involving Dedekind $\eta$ function and a special value for the complete elliptic integral of the first kind. I refrained from providing a complete ...
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2answers
276 views
How to evaluate sums in the form $\sum_{k=-\infty}^\infty e^{-\pi n k^2}$
Online, one may find the values of the following sums:
$$\sum_{k=-\infty}^\infty e^{-\pi k^2}=\frac{\pi^{1/4}}{\Gamma(3/4)}$$
$$\sum_{k=-\infty}^\infty e^{-2\pi k^2}=\frac{\pi^{1/4}(6+4\sqrt 2)^{1/4}}{...
4
votes
0answers
109 views
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.
$$ \frac{1}{(q)_\infty} \sum_{m \in \mathbb{Z}} q^{\frac{m^2}{2}} (-q^{-\frac{1}{2}}x)^m y^m(q^{1-m}y^{-1};q)_\infty
=...
5
votes
1answer
168 views
Did Gauss know Jacobi's four squares theorem?
This is a question that i have already asked on HSM stackexchange, and i decided to ask it again here because it's more mathematical than historic (to make a conclusion in this question one needs more ...
0
votes
1answer
80 views
Linearly equivalent divisors induce same projective embedding
Consider the torus $T=\mathbb C^n/\Lambda$ for some lattice $\Lambda$.
Say that two divisors $A,B$ on $T$ are linearly equivalent if their difference is the divisor of a meromorphic function on $T$. ...
1
vote
0answers
66 views
Can we extend the theta function $\theta(z)$ to p-adic numbers $\mathbb{Z}_p$?
Let $\theta(z) = \sum_{n \in \mathbb{Z}} q^{n^2}$ with $q = e^{2\pi i n z}$. Can we extend the theta function to $p$-adic arguments? Here's an example:
$$ \theta( 1 + p^k) = \sum_{n \in \mathbb{Z}^...
4
votes
0answers
55 views
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 ...
1
vote
0answers
57 views
No weight $2$ newforms on $\Gamma_0(4)$?
How could it be there are no modular forms of weight $2$ and level $4$. In the lmfdb database? We can easily think of one: $\theta(z)^4$ with
$\theta(z)=\sum_{n\in \mathbb{Z}} q^{n^2}$ . Where are ...
2
votes
1answer
75 views
$ \sum_{n=1}^{\infty} \frac{ x^{n^2} (1 + x^n) - x^n}{1 -x^n} = 0.$ ??
While studying theta functions I noticed
$$ \sum_{n=1}^{\infty} \frac{ x^{n^2} (1 + x^n) - x^n}{1 -x^n} = 0.$$
Why is that so ??
Is there a similar case with a term $x^{n^3}$ ??
1
vote
0answers
33 views
Theta notation for Strassen's multiplication
A student discovers a way to multiply 2×2 matrices using exactly 5 multiplications, instead of Strassen’s 7. What is the number M(n) of multiplications for the resulting algorithm to multiply n×n ...
