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55 views

Is there an analogue of Fourier series using infinite series of Jacobi elliptic functions to model doubly-periodic functions?

We can use Fourier series to model general singly-periodic functions. Infinite sums of orthogonal trigonometric polynomials can approximate any singly-periodic function that is continuous along its ...
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0answers
57 views

Jacobi's Elliptic functions - Kernel

I have a kernel expressed in terms of Jacobi's elliptic functions. In particular I have four functions $$\begin{eqnarray} &\rho_{11}(u,v)&=\frac{1- k sn u sn v}{sn u-k sn v},\nonumber\\ &\...
3
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1answer
81 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),...
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2answers
135 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 there were found ...
2
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1answer
175 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}...
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0answers
119 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) ...
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1answer
272 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
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0answers
146 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
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1answer
322 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
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1answer
216 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 ...
2
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1answer
187 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
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1answer
450 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
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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 ...
2
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2answers
549 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 $m,n \in \mathbb{Z}^2$. ...
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0answers
637 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
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3answers
514 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+...
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5answers
3k 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 ...