Questions tagged [theta-functions]
For questions about $\theta$ functions (special functions of several complex variables).
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Asymptotic of integral involving a theta function [closed]
I would appreciate some help with theta functions. Consider $\theta_3(u,q)$:
$$
\theta_3(u,q) = 1 + 2 \sum_{n = 1}^{+\infty} q^{n^2} \cos(2 n u)
$$
I am interested in the asymptotic of the following ...
- 1
1
vote
1
answer
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General form of Jacobi Theta Transformation $\sum_{n \in \mathbb{Z}} e^{n^2 \pi / x} = \sqrt{x} \sum_{n \in \mathbb{Z}} e^{n^2 \pi x} $
I was looking into the functional equation of $\zeta(s)$ and at one point the proof uses the Jacobi Theta Transformation:
$$\sum_{n \in \mathbb{Z}} e^{n^2 \pi / x} = \sqrt{x} \sum_{n \in \mathbb{Z}} ...
- 728
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Intersection of two quadrics is elliptic curve
I am currently studying Jacobi intersection models of elliptic curves, and so far I have found that whenever you have a lattice in $\mathbb{C}$ we have an embedding of $\mathbb{C}/\Lambda$ into $\...
1
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1
answer
27
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Definition of an integral function
I am new to theta functions, and reading about them in Elliptic Functions and Applications by Derek F. Lawden it is stated that these theta functions are "integral functions". What is the ...
- 547
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How to find the Lambert series expansion of this function
Let's consider the function
\begin{align}
\frac{\eta ^m\left( q \right)}{\eta \left( q^m \right)}
\end{align}
here η is the Dedekind eta function $
\eta \left( q \right) =q^{\frac{1}{24}}\prod_{n=1}^{\...
- 79
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vote
0
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maximal continuation of $\Pi_2(x)$
Consider the functions for $k\in \Bbb N$
$$ \Pi_k(x) := \sum_{n \in \Bbb N} e^{\frac{\log^k n}{\log x}} $$
$\Pi_1(x)$ converges for real $1/e<x<1$.
$\Pi_1(x)$ is a Riemann zeta function i.e. $\...
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Alternating series with Gaussian weights
I'm wondering if anyone knows anything about approximating alternating series with Gaussian weights, i.e., sums of the form
$$
\sum_{i=1}^{\infty}(-1)^{i+1}e^{-a(i-b)^2},
$$
for positive $a$ and $b$. ...
- 437
6
votes
3
answers
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How did Jacobi find his connection between theta functions and $q$?
I was 'reading' (I can't actually read german, but I can read math!) Jacobi's derivation of the ODE for $y(q) = \sum_{n=-\infty}^{\infty} q^{n^2} $.
On page 2 of the paper Jacobi states the following ...
- 17.5k
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Residue theorem and theta function identities
Let's use the classical definition
$$
\vartheta _1\left( z,q \right) =-i\sum_{n\in \mathbb{Z}}{\left( -1 \right) ^nq^{\left( n+\frac{1}{2} \right) ^2}e^{i\left( 2n+1 \right) z}}\,\,\,\,\,\
q=e^{i\pi \...
- 79
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votes
1
answer
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How to prove the equality of power series below?
Assume
$$
F\left( x \right) := \sum_{n=-\infty}^{+\infty}{x^{\left( n+\frac{1}{2} \right) ^2}}, G\left( x \right) := \sum_{n=-\infty}^{+\infty}{x^{n^2}}, H\left( x \right) := \sum_{n=-\infty}^{+\infty}...
- 194
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What is the transformation formula of Theta series associated the quadratic form over totally real field?
I am currently reading the book of Erich Hecke "Lectures on the Theory of Algebraic Numbers", and it is not clear for me to understand his notation of theta series associated to quadratic ...
- 277
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votes
1
answer
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Mistake computing $\sum_{n=1}^{+\infty} \frac{n}{e^{2\pi n}-1} = \frac{1}{24}-\frac{1}{8\pi}$
I recently gave a try to show that
$$\sum_{n=1}^{+\infty} \frac{n}{e^{2\pi n}-1}=\frac{1}{24}-\frac{1}{8\pi} $$
without using the Theta function or Mellin transform, but I ended up with twice the ...
- 167
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votes
1
answer
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Understanding the proof of Theorem 10.1 in Montgomery & Vaughan's Multiplicative Number Theory
In the last step of the proof of Theorem 10.1 in the book Multiplicative number theory I: Classical theory by Hugh L. Montgomery, Robert C. Vaughan I couldn't understand what exactly "turn the ...
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How does Jacobi form lives projectively on the torus $\mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})$?
I saw the statement in the question from the book Moonshine Beyond the Monster.
We are given the definition : a group hom $\rho : G \rightarrow PGL(V)$ is called a projective representation.
I can't ...
- 477
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Close form expression for an integral with z derivative of jacobi theta function
I have an expression of the form
$$
\tag{1}
A(\chi) = \int_{0}^\infty\sum_{i=0}^\infty (-1)^{i+1}\frac{(2i+1)\chi}{\sqrt{t}}\exp\left(\frac{-(2i+1)^2\chi^2}{t}\right)\mathrm{d}t.
$$
If I am not ...
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