Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [theta-functions]

For questions about $\theta$ functions (special functions of several complex variables).

In mathematics, $\theta$ functions are special functions of several complex variables. They are important in many areas, including the theories of abelian varieties and moduli spaces, and of quadratic forms. They have also been applied to soliton theory. When generalized to a Grassmann algebra, they also appear in quantum field theory. The most common form of $\theta$ function is that occurring in the theory of elliptic functions.

338 questions
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What is a Theta Function?

What exactly is a theta function $$\vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp (\pi i n^2 \tau + 2 \pi i n z)= 1 + 2 \sum_{n=1}^\infty \left(e^{\pi i\tau}\right)^{n^2} \cos(2\pi n z) = \sum_{n=-\infty}^\infty q^{n^2}\eta^n"$$ and how does it…
bolbteppa
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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 exponential function satisfies? If so, which…
aleden
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How to compute $\sum_{n=0}^{\infty} \frac{(-1)^n}{\cosh((n+\frac{1}{2}) \pi)}$

How to compute $\sum_{n=0}^{\infty} \frac{(-1)^n}{\cosh((n+\frac{1}{2}) \pi)}$? My attempt was trying to consider the Mellin transform of $\frac{1}{\cosh(x)}$ and use the inverse Mellin transform as a representation of…
Dqrksun
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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 & 0\\ 0 & 2 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0\\ -1…
Xiang Zhao
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4
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1 answer

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 $$\begin{eqnarray} \vartheta(\tau,…
user581023
4
votes
2 answers

Jacobi Theta function inequality

I am trying to show that $$\sum_{n=1}^\infty e^{- \pi n^2 x} < \frac{1}{2} x^{-\frac{1}{2}}, \; \forall x>1$$ Here's what I'm doing: $$\sum_{n=1}^\infty e^{- \pi n^2 x} < \sum_{n=1}^\infty e^{- \pi n x} = \sum_{n=0}^\infty e^{- \pi n…
Nicolo Canali De Rossi
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1 answer

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 $\sigma^2 > 0$. Then $$\sum_{n \in \mathbb{Z}}…
Thomas
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Closed-form of an integral involving a Jacobi theta function, $ \int_0^{t} \theta_3(e^{-\pi^2 (t-\tau)}) \, \theta_2(e^{-\pi^2 \tau}) \ d\tau =1$

Numerical calculation of a Duhamel-Integral coming up considering a unsteady state diffusion in a thin film electrode with zero initial concentration leads to the following strange identity: $$ \int_0^t \theta_3(e^{-\pi^2 (t-\tau)}) \,…
stocha
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theta function like boundary conditions

I have the boundary conditions: $$f_\alpha(z+1)=\mathrm{e}^{-\pi N(2z-1)}f_\alpha(z)$$ and $$f_\alpha(z+i)=f_\alpha(z),$$ now, I know a theta function of the form $$f_\alpha(z|\tau)=\sum_{m\in\mathbb{Z}}\mathrm{e}^{i\pi(m+\alpha/N)^2N\tau+2\pi…
Lewis Proctor
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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 matrices? Use Θ-notation to express your estimate.
Ashley Winters
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Is the $z\gg 1$ behavior of the theta function $\theta_1(z;q)$ known?

Is the $z\gg 1$ behavior of the theta function $\theta_1(z;q)$ known? It seems naively like it could be estimated by a Gaussian integral which would give an answer along the lines of $e^{bz^2}$ for some $b$. I have had no luck finding the answer…
user245692
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Ratio of theta function derivatives with theta function

I have the following ratios I want to compute. $$ \frac{ \left( \frac{\partial \vartheta_3(v, q)}{\partial v} \right)^2 }{C + \left(\vartheta_3(v, q)\right)^2 }, $$ where $C$ is a constant. $$ \frac{ \left( \frac{\partial \vartheta_3(v, q)}{\partial…
CfourPiO
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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.
Mona
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