# Questions tagged [elliptic-functions]

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### Resources on the stationary Schrödinger equation with the soliton potential

I am studying the following Lamé equation in the Jacobi form \begin{equation} -\frac{d^2 v}{dx^2} - \left(2k^2 \operatorname{cn}^{2}\left [x\;|\;k\right ]\right)v = \lambda v, \end{equation} ...
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### double periodic functions with real variables

$u(x+X,y)=e^{iky}u(x,y)\\ u(x,y+Y)=u(x,y)$ This is a quasi double periodic boundary condition. x and y are real numbers. I'm wondering whether there exists a general formula of real variable ...
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### How to determine the closed form of this Fourier series?

Consider the Series $$S(z) \equiv \sum_{n \in \mathbb{Z}, n \ne 0} \frac{ 1 }{ \sin n\pi \tau \sin 2n \pi \tau } e^{2\pi i n z} \ , \quad \operatorname{Im}\tau > 0$$ I am trying to find its ...
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### Infinite sum of iterated integrals of matrix products

Originally asked over at Stackexchange (https://math.stackexchange.com/questions/4169812/infinite-sum-of-iterated-integrals-of-matrix-products), but this forum was deemed more appropriate. The problem:...
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1 vote
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### Algebraic relation amongst an elliptic function and its convolution

NOTE: I edited this question, following the comments of Alexander Eremenko and Paul Garrett. I have a question concerning elliptic functions that maybe you can help me shed light on. I am a ...
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1 vote
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### Second order ODE with Jacobi elliptic function coefficients

I posted this last week on the Mathematics Stack Exchange but have not been able to get an answer. I have read the rules and couldn't find anything against this, but please remove this question if ...
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### Are these 5 the only eta quotients that parameterize $x^2+y^2 = 1$?

Given the Dedekind eta function $\eta(\tau)$, define, $$\alpha(\tau) =\frac{\sqrt2\,\eta(\tau)\,\eta^2(4\tau)}{\eta^3(2\tau)}$$ $$\beta(\tau) =\frac{\eta^2(\tau)\,\eta(4\tau)}{\eta^3(2\tau)}\quad\;$$...
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### Evaluating a Fermi gas problem for a SO(2N+1) matrix integral

I have the following multiple integral derived from a random matrix calculation I wish to evaluate $$\int_0^{\pi} dx_1 dx_2 \cdots dx_n \rho(x_1,x_2)\cdots \rho(x_n,x_1)$$ where the $\rho$ functions ...
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### Behaviour of elliptic functions near degenerate lattice

What can be said about elliptic functions (Weierstrass $\wp$, $\sigma$, Jacobi $\theta$, sn, cn, etc.) in the limit of degenerate lattice. By "degenerate" I mean $\tau = \omega_3/\omega_1$ tends to a ...
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### Solution of an equation with Jacobi theta function

I have been struggling with this equation for some time and I do not seem to find any conclusive answer (it's from my research, not a homework). It has to do with the real solutions $x$ to the ...
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### Bounding an elliptic-type integral

Let $K>L>0$. I would like to find a good upper bound for the integral $$\int_0^L \sqrt{x \left(1 + \frac{1}{K-x}\right)} \,dx.$$ An explicit expression for the antiderivative would have to ...
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### Jacobi and Weierstrass elliptic function

Jacobi elliptic function $\mathrm{sn}$ is defined as $$\operatorname{sn}(u,k)=x\Leftrightarrow u=\int_0^x \frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}.$$ and Weierstrass sigma function $\sigma$ is defined as ...
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### Special values of the modular J invariant

A special value: $$J\big(i\sqrt{6}\;\big) = \frac{(14+9\sqrt{2}\;)^3\;(2-\sqrt{2}\;)}{4} \tag{1}$$ I wrote $J(\tau) = j(\tau)/1728$. How up-to-date is the Wikipedia listing of known special values ...
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### Jacobi's elliptic functions and plane sections of a torus

In $\mathbb R^3$ with Cartesian coordinates $(x,y,z),$ revolve the circle $(x-\sqrt 2)^2+z^2 =1,\ y=0$ about the $z$-axis. This yields a torus embedded in $3$-space that is conformally equivalent to ...
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### Elliptic curve with CM by $(1+\sqrt{-11}) /2$

Can someone explain to me on how to obtain the endomorphism for elliptic curve with CM by $(1+\sqrt{-11}) /2$? Given the elliptic curve over $F_{p}$ as $y^2=x^3-13824/539 x + 27648/539 \dots$ how do ...