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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247 views

Can a doubly periodic function be locally univalent?

I am looking for a meromorphic doubly periodic function such that the function is locally univalent. A standard meromorphic doubly periodic funtion is the Weirestrass $\wp$ function, defined as $$\wp(...
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316 views

How to work out this elliptic function?

Let $f(x) = \sum\limits_{(n,m)\in\mathbb{Z}^2} \frac{1}{(x+ n + i m )^2}$ If feel it should be $1/E(x)$ where $E$ is some elliptic function, like $sn^2$. But Wolfram Alpha is giving me some strange ...
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Estimatives for elliptic systems involving the laplacian

Considering the problem \begin{equation} \left\{ \begin{array}[c]{11} \Delta(\Delta \chi -\chi) = 0 & \text{in } \Omega, \\ \Delta \chi -\chi = h_2 - h_1, & \text{on } \partial\Omega \\ \end{...
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3 votes
1 answer
223 views

Generating function of the product of Legendre polynomials

The generating function of the product of Legendre polynomials for the same $n$ is given by \begin{aligned} \sum_{n=0}^{\infty} z^{n} \mathrm{P}_{n}(\cos \alpha) \mathrm{P}_{n}(\cos \beta)&=\frac{\...
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6 votes
2 answers
465 views

Can the theory of elliptic functions developed from purely geometric considerations?

I always had this question, but was unable to get a definitive answer to it. There is the theorem of division of the arc length of the lemniscate with ruler and compass. So I always wondered, is it ...
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1 answer
206 views

Regarding the Weierstrass $\wp$-function of the hexagonal lattice

Playing with the Weierstrass $\wp$-function of the hexagonal (or triangular) lattice $\mathbb{T}$, $$ \wp'(z)^2 = 4 \wp(z)^3 - 1, $$ I noticed that the zeros of $\wp'(z) + \sqrt{3}$ are $$ \frac{\...
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0 answers
51 views

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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0 answers
128 views

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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260 views

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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3 votes
2 answers
344 views

About the integral $\int_{0}^{1}\frac{\log(x)}{\sqrt{1+x^{4}}}dx$ and elliptic functions

NOTE: I post this question on math.stackexchange but nobody answered, so I try here. For a work we need to evaluate the following integral $$\int_{0}^{1}\frac{\log\left(x\right)}{\sqrt{1+x^{4}}}dx=\,-...
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1 vote
1 answer
103 views

Analytic function with q- difference equation involving theta

Consider the analytic space $\mathbb{C}^{*}$ with coordinate $z$. Let $q$ be some parameter with $|q|<1$ and define the analytic function $$\theta(z;q):=\sum_{n\in\mathbb{Z}}q^{\binom{n}{2}}(-z)^{n}...
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1 vote
0 answers
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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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The loss of double periodicity (ellipticity)

Consider a meromorphic function $f(\mathfrak{a}_1, \mathfrak{a}_2)$, such that $$ \begin{align} f(\mathfrak{a}_1, \mathfrak{a}_2) = f (\mathfrak{a}_1 + 1, \mathfrak{a}_2) = f(\mathfrak{a}_1 + \tau, \...
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Is there some precise way in which modular cocycle $c/(c\tau +d)$ "is" the generator of $H^1(\mathcal{M}_{ell}, \omega^2) \cong \mathbb{Z}/12$?

It is semiclassical that the extension class $\text{Ext}^1(\omega^{-1},\omega)\cong H^1(\mathcal{M}_{ell},\omega^2)$ over the modular stack $\mathcal{M}_{ell}/\mathbb{Z}$ is nonvanishing, being cyclic ...
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2 votes
1 answer
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Index and weight of Weierstrass $\sigma$-function as a Jacobi form, versus a statement in a note by Zagier

Let $$\sigma_L(w; \tau):=\frac{w}{\exp\left(\sum_{k\ge 2} 2G_k(q)\frac{w^k}{k!}\right)}$$ be the version of the Weierstrass $\sigma$-function which is used to orient $\text{tmf}$; here $w=2\pi i z$, $...
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3 votes
1 answer
193 views

Understanding the implementation of the $p$-adic(?) sigma function in SageMath

I'm trying to understand the (pretty undocumented) method .sigma() method for formal groups of elliptic curves, as listed here. The source code looks like this: <...
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10 votes
0 answers
309 views

Is this elliptic integral identity known?

Thinking about some physical problem, I came across the following identity: $$\phi^2\Pi\left(-\phi,\frac{1}{\sqrt{2}}\right)+\phi^{-2}\Pi\left(\phi^{-1},\frac{1}{\sqrt{2}}\right)=\frac{\pi}{\sqrt{2}}+...
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1 vote
0 answers
44 views

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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3 votes
1 answer
173 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{...
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1 vote
0 answers
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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 $...
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5 votes
1 answer
285 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_{...
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3 votes
1 answer
621 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^...
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1 vote
0 answers
73 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 ...
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4 votes
0 answers
214 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 $\...
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1 vote
1 answer
313 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-...
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15 votes
4 answers
1k 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 ...
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7 votes
1 answer
458 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(...
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7 votes
2 answers
406 views

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

Given the Dedekind eta function $\eta(\tau)$, define, $$\alpha(\tau) =\frac{\sqrt2\,\eta(\tau)\,\eta^2(4\tau)}{\eta^3(2\tau)}$$ $$\beta(\tau) =\frac{\eta^2(\tau)\,\eta(4\tau)}{\eta^3(2\tau)}\quad\;$$...
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3 votes
0 answers
93 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 ...
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2 votes
1 answer
84 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 ...
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6 votes
2 answers
504 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 ...
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8 votes
2 answers
285 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 ...
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2 votes
1 answer
87 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 $$ ...
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-1 votes
1 answer
400 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^{...
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2 votes
0 answers
61 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 ...
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23 votes
5 answers
3k 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 ...
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4 votes
2 answers
335 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 ...
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-2 votes
2 answers
383 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)}}\...
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3 votes
1 answer
443 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 ...
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9 votes
4 answers
966 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 ...
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10 votes
2 answers
1k 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 ...
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3 votes
1 answer
253 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 ...
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2 votes
1 answer
257 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}{\...
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2 votes
0 answers
236 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{...
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3 votes
1 answer
169 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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5 votes
2 answers
222 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 ...
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3 votes
1 answer
344 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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3 votes
0 answers
568 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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1 vote
1 answer
893 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 ...
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