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Questions tagged [elliptic-functions]

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5
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1answer
115 views

What are the modularity properties of Weierstrass sigma function?

I'm a little confused at the sigma orientation of tmf, see e.g. Witten genus and its references. The Weierstrass sigma function can be written as $$\sigma_L(z)(q)=\frac{z}{\exp\left(\sum_{k\ge 2} G_k(...
0
votes
0answers
57 views

Evaluate a pair of integrals involving dilogarithms over the unit interval

These are two variations on the "Bonus round" problem, expertly address by student at the end of his answer to A pair of integrals involving square roots and inverse trigonometric functions over the ...
6
votes
2answers
339 views

Are these 5 the only eta quotients that parameterize $x^2+y^2 = 1$?

Given the Dedekind eta function $\eta(\tau)$, define, $$\alpha(\tau) =\frac{\sqrt2\,\eta(\tau)\,\eta^2(4\tau)}{\eta^3(2\tau)}$$ $$\beta(\tau) =\frac{\eta^2(\tau)\,\eta(4\tau)}{\eta^3(2\tau)}\quad\;$$...
3
votes
0answers
86 views

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 ...
0
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0answers
53 views

Rewriting an elliptic integral in terms of theta functions

I wish to demonstrate the following from arXiv:hep-th/9808043v2 (equations (3.2) to (3.4)). This is rewriting the following incomplete elliptic integral of the third kind $ \Phi_{\tilde{h}}(h) = \...
2
votes
1answer
68 views

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 ...
6
votes
2answers
373 views

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 ...
8
votes
2answers
253 views

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 ...
2
votes
1answer
79 views

Limits of a quasiperiodic function with two pseudoperiods

Let $\beta$ be a real number such that $\beta^2\notin\mathbb{Q}$. For any smooth function $f$ on $\mathbb{R}$ that decreases sufficiently at infinity, for example a Gaussian function, let us define $$ ...
-1
votes
1answer
326 views

Infinite sum and product associated with the Weierstrass elliptic function [closed]

Can anyone help me figure out how the identity below was obtained? $ \frac{1}{\sqrt{(e_1-e_3)(e_2-e_3)}} = R \prod \limits_{n=1}^{\infty} \left(1 - \frac{1}{R^{4n}} \right)^{-4}\left(1 + \frac{1}{R^{...
2
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0answers
53 views

Will a slightly differently shaped torus make this guess about plane sections of a torus true?

Jacobi's elliptic functions and plane sections of a torus After Greg Egan posted an excellent answer to a question of mine, which I accepted, I posted my own answer, linked above. The question ...
20
votes
5answers
2k views

The letter $\wp$; Name & origin?

Do you think the letter $\wp$ has a name? It may depend on community - the language, region, speciality, etc, so if you don't mind, please be specific about yours. (Mainly I'd like to know the English ...
4
votes
2answers
227 views

Differentiating the inverse Weierstrass P-function

I will begin with some background: The solutions $\theta$ of $$\cos \theta=x $$ constitute of two families, each of which is an arithmetic progression. Namely, if $\arccos x$ denotes any particular ...
-2
votes
2answers
361 views

Expression for infinite product

can anyone show me how $$\displaystyle\frac{4}{R}\displaystyle\Pi_{n=1}^{\infty} \left(\frac{1+R^{-4n}}{1+R^{-4n+2}}\right)^4= \frac{1}{R}\left(1+2 \sum_{n=1}^ {\infty} \frac{1}{R^{2n(n+1)}}\...
3
votes
1answer
341 views

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 ...
4
votes
2answers
331 views

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 ...
8
votes
2answers
771 views

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 ...
3
votes
1answer
237 views

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 ...
2
votes
1answer
231 views

Identity from Lectures on the Theory of Elliptic Functions by Harris Hancock

I'm working on Example $4$, page $262$, of Harris Hancock's book Lectures on the Theory of Elliptic Functions which reads: Prove that $\dfrac{1}{\operatorname{sn}(iu,k)^2} + \dfrac{1}{\...
2
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0answers
221 views

Jacobi's Elliptic functions - Kernel

I have an integral equation with a kernel expressed in terms of Jacobi's elliptic functions. In particular I want to solve the following equation: $$\lambda \begin{pmatrix} X_1(u) \\ X_2(u) \end{...
3
votes
1answer
132 views

Functional equations associated with addition theorems for elliptic functions

I'm trying to read the article "Functional equations associated with addition theorems for elliptic functions and two-valued algebraic groups" by Bukhshtaber,V. M. Russian Mathematical Surveys(1990),...
5
votes
2answers
193 views

Are the Gessel sequence integers composite for all $n\ge 3$?

The Gessel sequence is known for Ira Gessel's Lattice Path Conjecture of $2001$, which has been proved by Kauers, Koutschan and Zeilberger in $2009$ with the aid of a computer. Later, other proofs ...
3
votes
1answer
270 views

How can one parametrize a real elliptic normal curve such that four points are coplanar iff their parameters sum to zero?

Let $E \subset \mathbb{P}^3_{\mathbb{R}}$ be a real elliptic normal curve with two non-null-homotopic connected components. Is there a parametrization $$ \chi: (\mathbb{R}/\mathbb{Z})\times (\mathbb{Z}...
3
votes
0answers
449 views

Integration of Weierstrass elliptic functions

Is there a way to integrate the following expression $$ \int \frac{dt}{\cal{P}(t;g_2,g_3)-c} $$ where $\cal P$ is the Weierstrass elliptic functions and $g_2$, $g_3$, and $c$ are some (real) ...
1
vote
1answer
573 views

Hurwitz, A. and R. Courant: Funktionentheorie , elliptic functions part

Can some one suggests an English text covering that part of the book dealing with elliptic functions. As i understand from here, there is no translation of the full book to English but maybe another ...
3
votes
0answers
185 views

Weierstrass's elliptic function-type zeta function

What is known about the following Weierstrass's elliptic function-type zeta function $\sum_{m,n \in \mathbb{Z}} \frac{1}{(z+m+n\tau)^s}$, for $z \in \mathbb{C} \backslash \mathbb{Z} + \tau \mathbb{Z}...
3
votes
1answer
450 views

Elliptic units and Euler system

Maybe this question is quite obscure and ambiguous. I am really sorry for such ambiguity. My question is, what is the good thing we get from defining elliptic units and Euler system? There are lots ...
1
vote
1answer
227 views

A proof of energy functional appearing in the regularity of elliptic and parabolic equations

I have got trapped in this problem for nearly two years when I dealt with regularity of solutions of elliptic and parabolic equations. I have not found a nice proof to support this assertion. Now I am ...
3
votes
1answer
387 views

Are traditional notations for elliptic integrals/functions in Latin or Greek letters?

I am doing some calculation involving elliptic integrals/functions, and find the notations confusing. In Wittaker-Watson, the "Jacobi's earlier notation" H(u) is called the Eta-function, so the "H" ...
6
votes
1answer
698 views

Evaluating the average distance from a point in the unit disk to the disk

I am interested in finding the average Euclidean distance from a point $(x,y)\in\mathbb{D}_2$, the unit disk $\{(u,v):u^2+v^2\leq 1\}\subseteq\mathbb{R}^2$, to the disk $\mathbb{D}_2$. This amounts to ...
26
votes
2answers
2k views

Mathematician, Graciano Ricalde

Does anyone understand more precisely how to explain the 5th degree equation and elliptic functions accomplishments of Mathematician Graciano Ricalde? I am his great grand-daughter and trying to ...
3
votes
2answers
613 views

Special values of a doubly periodic meromorphic function

Consider the following function: $G(z) = \prod_{n \in \mathbb{Z}} {1 \over{\tanh^2\left(\pi\left(z-n\right)\right)}}$. By constuction, it has poles at $z=m+in$ with $m,n \in \mathbb{Z}^2$. ...
10
votes
0answers
731 views

Convexity of Jacobi's theta function with zero argument

This question may be elementary, I have asked it on math.stackexchange.com but have not received any answer yet. Note that I am not an expert on theta/elliptic functions. Define Jacobi's theta ...
4
votes
3answers
577 views

question about the recursion equation: $x_{n+1}x_{n−1}=x_n^2(1−4x_n)$

Hi all, I am trying to slove the recursion equation: $x_{n+1}x_{n−1}=x_n^2(1−4x_n)$ in the form of $x_n=x_n(x_1,x_2)$ or $x_n=x_n(c_1,c_2)$, and finally get the limit of the ratio: $\dfrac{x_n}{x_{n+...
19
votes
5answers
4k views

Proofs of Jacobi's four-square theorem

What are the nicest proofs of Jacobi’s four-square theorem you know? How much can they be streamlined? How are they related to each other? I know of essentially three aproaches. Modular forms, as in,...