# Questions tagged [elliptic-functions]

The elliptic-functions tag has no usage guidance.

53
questions

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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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155 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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268 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**answer

90 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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89 views

### 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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72 views

### 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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105 views

### 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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**1**answer

117 views

### 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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150 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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298 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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34 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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**1**answer

118 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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76 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 $...

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**1**answer

258 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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**1**answer

331 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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67 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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178 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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244 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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**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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**1**answer

372 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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394 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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91 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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**1**answer

79 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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**2**answers

448 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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280 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**

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**1**answer

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
$$
...

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**1**answer

376 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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58 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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**5**answers

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 ...

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289 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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379 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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**1**answer

405 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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786 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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960 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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**1**answer

247 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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**1**answer

246 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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**0**answers

231 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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**1**answer

146 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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213 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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**1**answer

300 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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524 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**answer

781 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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204 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}...

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**1**answer

545 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 ...

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**1**answer

234 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 ...

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**1**answer

538 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" ...

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**1**answer

818 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 ...

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**2**answers

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 ...

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654 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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769 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 ...