Questions tagged [elliptic-functions]

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3
votes
1answer
92 views

Conformal mapping between two right-angled triangles

I want to derive a conformal mapping $f\!:\!A\!\to\! B$ where $A=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq x \}$ and $B=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq \frac{...
1
vote
0answers
72 views

Snoidal wave solutions of the $\phi^4$ model

I want to prove the existence of snoidal wave solutions of the $\phi^4$ model, given by $$u_{tt}-u_{xx}=u-u^3,\; (x,t) \in \mathbb{R}\times \mathbb{R}.$$ So, we are looking for solutions in the form $...
5
votes
1answer
247 views

How to prove some identities about infinite product?

Recently, I read one paper titled Modular equations and approximations to π by Ramanujan, in which there are some formulas for $q=\pi i \tau$( where $\tau=x+yi, y>0$, hence $|q|<1)$ : $$\prod_{...
3
votes
1answer
111 views

Inverse of the incomplete elliptic integral of the second kind

The incomplete elliptic integral of the second kind $E(\varphi \, | \,k)$ is defined as follows: $$E(\varphi \, | \,k) = \int_0^\varphi \sqrt{1-k^2\sin^2\theta} \, \mathrm{d}\theta $$ Where $0<k^...
0
votes
0answers
40 views

Constructing a Meromorphic function on a genus 1 surface with prescriped divisor using Jacobi elliptic functions

I need to construct a meromorphic function $f \in \mathcal{M}(X)$ using the Jacobi elliptic functions on a genus $1$ surface with divisor of the form: $(f) = n \cdot P_0 + P_1 - n \cdot Q_0 - Q_1$, ...
1
vote
0answers
54 views

Construct a doubly periodic function on $\mathbb{C}$ using Jacobi elliptic functions with anti-holomorphic involution

I would like to find explicit examples of non-constant meromorphic doubly periodic $f(z)$ on $\mathbb{C}$, i.e. a meromorphic function on $\mathbb{C} / \Gamma$ for some lattice $\Gamma$, such that ...
4
votes
0answers
132 views

Expressing the inverse Dixon function in terms of more familiar functions

If $x^3+y^3-3\alpha xy=1$, is there an expression for the integral $$\int_0^z \frac{\mathrm dx}{y^2-\alpha x}$$ in terms of more familiar functions? A.C. Dixon introduced the elliptic functions $\...
0
votes
0answers
38 views

Identity relation on Theta functions on 5 variables

I stumbled in an equation on the theta functions that is probably false itself, but promoted me a question. Let's define the notation $$\theta_{p}\left(0\right)\theta_{q}\left(z_{1}\right)\theta_{r}\...
1
vote
1answer
172 views

Weierstrass elliptic function in Laurent series form [closed]

Could anyone help me to figure out how $$ f_0(z) = \wp (\log z; i \pi, \log \rho) $$ where $\wp$ denotes the Weierstrass elliptic function and $i \pi$, $\log \rho$ are its half-...
8
votes
3answers
810 views

Determination of special values of Eisenstein series

We have the Eisenstein series of weight $k$: $G_k(z)=\frac 1 2 \sum_{m,n} \frac 1 {(mz+n)^k}$. Can we evaluate it in closed form for some special values of $z$, eg. $z=i$ or $z=\omega$? It is clear by ...
5
votes
1answer
200 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(...
7
votes
2answers
367 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
89 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 ...
2
votes
1answer
77 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
410 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
272 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
83 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
352 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
votes
0answers
57 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 ...
21
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
265 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
377 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
369 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 ...
6
votes
3answers
574 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
862 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
242 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
242 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
votes
0answers
226 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
135 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
200 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
291 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
499 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
670 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
195 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
505 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
232 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
471 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
734 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
630 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
749 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
579 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+...
20
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,...