Questions tagged [elliptic-functions]

Questions on doubly periodic functions on the complex plane such as Jacobi and Weierstrass elliptic functions.

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Real analytic elliptic functions

First off, I'm sorry that the question is somewhat open-ended. I am interested in functions such as $$f(z;\tau)=\sum_{m,n\in\mathbb{Z}}\frac{1}{(|z+m\tau+n|^2+a^2)^{\frac{3}{2}}}\,,\quad g(z;\tau)=\...
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Integrating a rational fraction with Jacobi elliptic funtion

I am trying to compute integrals of rational fractions involving Jacobi elliptic functions in real case. The initial integral has the following form where all coefficients are strictly positive: $$ \...
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Is the Weierstrass $\wp$-function compatible with automorphisms of $\mathbb{C}$?

Let $\Lambda$ be a lattice in $\mathbb{C}$ and $\sigma\in\mathrm{Aut}(\mathbb{C})$. Is it true that \begin{equation} \tag{$*$} \sigma(\wp(z; \Lambda))=\wp(\sigma(z); \sigma(\Lambda))? \end{equation} ...
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Birationally equivalent elliptic curves and singularities

I got the following cubic elliptic curve from some physical problem $$E_c(\mathbb{C}): w^2=4 z^3-zG_2-G_3,$$ where $G_2=3 \alpha ^2+\gamma$ and $G_3=\alpha ^3-\alpha \gamma -\beta ^2$ for known ...
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Complete integral involving square of Jacobi elliptic functions

I've come across the following integral while studying the motion of a steady 2D fluid with stream function cos(x)cos(y), the flow of which involves Jacobi elliptic functions [sn, cn, dn]. I'm hoping ...
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6 votes
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Closed form for definite integrals invovling Jacobi elliptic functions

In a 1879 work, Glaisher proves the following closed forms $$\int_{0}^{K\left(k\right)}\log\left(\text{sn}\left(z;k\right)\right)dz=-\frac{1}{4}\pi K^{\prime}\left(k\right)-\frac{1}{2}K\left(k\right)\...
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Exactness of the volume form on torus minus a point

It follows from $H^2(T^2,\mathbb{Z}) = \mathbb{Z}$ that the volume form on the torus $$ v = \frac{i}{2} dz \wedge d\bar{z} $$ cannot be exact. Here I use complex coordinate $z$ and $\bar{z}$ and think ...
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Jacobi Elliptic function in terms of theta functions

I am new to using these functions and am confused about what is a function of what. If I want to solve $sn(x,k)$ for a given x, and use this equation: $sn(u,k)=\frac{\theta_{2}}{\theta _{3}}\frac{\...
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5 votes
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Is this function an alternative solution to the nonlinear pendulum?

Is this function an alternative solution to the nonlinear pendulum? Introduction I am working with the differential equation of the frictionless nonlinear pendulum: $$\ddot{\theta}(t) + b\,\sin(\theta(...
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Biharmonic problem with boundary conditions on Laplacian

Let the problem, where $\Omega$ is an open set of $\mathbb{R}^3$ and $h_1$ and $h_2$ are regular given functions \begin{equation}\nonumber%\label{eq:Pe}\tag{$P_{\varepsilon}$} \left\{ \begin{array}[c]...
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How to approach the problem of summation of Eisenstein series on shifted lattices?

This question is an attempt to complete the issues discussed in a previous question of mine (How did Gauss sum Eisenstein series?), since my updated question did not recieve any attention. In my ...
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An Elliptic Curve has 3 real roots if and only if $\frac{\omega_2}{\omega_1}$ is purely imaginary

Define $\Lambda=\{n\omega_1+m\omega_2 \mid n,m\in \mathbb{Z}\}$ for $\omega_1, \omega_2\in \mathbb{C}$ and $\wp(z)$ is the corresponding Weierstrass elliptic function. I want to show that $e_1=\wp(\...
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Relation Between Jacobi's Theta Function and Weierstrass $\wp$ Function

I am reading Elliptic Curves by Moll and McKean and it defines Jacobi's theta function on the lattice $\Gamma = \{n+m\omega \mid m,n \in \mathbb{Z}\}$ for a $\omega \in \mathbb{H}$ as below: $$\...
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Trying to find an elliptic function such that $\lim_{z \rightarrow \lambda} \frac{f'(z)}{f(z)} = x \;\forall \lambda \in \Lambda$

Let $\Lambda := \{m + in : m,n ∈ \mathbb{Z}\} \subset \mathbb{C}$ and $x \in \mathbb{R}$. I am trying to find an elliptic function $f : \mathbb{C} \setminus \Lambda \rightarrow \mathbb{C}$ such that $\...
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Relation Between Involutions on an Elliptic Curve and the Corresponding Complex Torus

This is a question from the book Elliptic Curves by Henry McKean and Victor Moll: Consider the cubic $X_1: y^2=x^3-x$. It admits the involution $(x,y) \mapsto (\frac{-1}{x}, \frac{y}{x^2})$. It is ...
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Finding a minimun by using ellipticity (optimization methods)

I would be grateful if someone can check my claims or recommend another way of solution: Let $n \in \mathbb{N}^{*}(n \neq 0)$, $A$ a real square matrix, and $\mathbf{c}$ a vector in $\mathbb{R}^{n}$. ...
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Weierstrass invariant on a particular lattice

Consider the function \begin{eqnarray*} \wp(z)=\frac{1}{z^2}+\sum_{a\in \Lambda^*}\left( \frac{1}{(z-a)^2}-\frac{1}{a^2}\right) \end{eqnarray*} where $\Lambda = \mathbb{Z}\bigoplus\omega\mathbb{Z} $ ...
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Integral of Elliptic Function giving Period

I am working on the following problem that was previously posted regarding the integral of an elliptic function. I understand the entire solution that was posted; however, there is a small piece of ...
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Argument Principle for the elliptic function

Suppose that $a_1, \ldots, a_r$ and $b_1, \ldots, b_r$ are the zeros and poles, respectively, in the fundamental parallelogram of an elliptic function $f$. Show that $$a_1+\cdots +a_r-b_1-\cdots -b_r=...
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Is the solution to the Weierstrass ODE over the reals still the Weierstrass elliptic function?

Suppose that we have a function $f(x)$ of a real variable $x$ in some domain of $U\subseteq\mathbb R$. Also, suppose that this function satisfies the ODE $$(f'(x))^2=4f^3(x)-g_2f(x)-g_3\,,$$ which is ...
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Are subnormal and polar subnormal equal?

I have been studying Jacobi Elliptic functions and I am trying to understand the proof that the curve which ellipse describes is of the form $dn(x)$.I found this proof: Where there is written $MG = - ...
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13 votes
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How did Gauss sum Eisenstein series?

Entry 61 in Gauss's diary states (this is a translation from latin): From the integer powers of $$\int_0^1 \frac{dx}{\sqrt{1-x^4}}$$ depends $$\sum_{m,n}(\frac{m^4 - 6m^2n^2 +n^4}{(m^2+n^2)^4})^k$$. ...
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Jacobi elliption function values for complex-valued modulus

I have a program that can compute Jacobi elliptic function values for real-valued argument u and modulus m. I would now like to ...
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Question about addition of points on elliptic curves in relation to elliptic integrals

I'm working on a long form piece on the development of elliptic curve cryptography, basically tracing the use of a point at infinity back to the discovery of the rules for linear perspective in the ...
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Normalized periods of elliptic functions

I am a bit confused, and inexperienced. I have an elliptic function with periods $2\omega_1,2\omega_2$. I heard one can normalize these periods as $\tau=\frac{\omega_2}{\omega_1}$. I know, you can ...
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Weierstrass elliptic function differential form

this question should be easy to answer yet I can't find the solution. I want to see how the following equation is true, when $\wp(z)$ is the Weierstrass elliptic function: $$\wp'(z)^2=4\wp(z)^3-g_2\wp(...
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Alternative way of showing the modular transformation property of $G_2^*(z)$

I'm currently reading D. Zagiers elliptic modular forms and their application. And like many others in order to prove the modular transformation property of $G_2^*(z)$ he refers to the form $G_2^*(z) =...
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Relation between Weierstrass $ \wp $ function and elliptic curve

I want to see how the Weierstrass $ \wp $ function and its derivative can define the curve $$ y ^ 2 = 4x^3 - g_2 x - g_3 $$ I think I can simulate the $ \wp $ function and its derivative properly, but ...
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What is the significance of Gauss-Weierstrass's derivation of "Al functions"?

In a fragment entitled "inversion of the elliptic integral of the first genus" (Gauss's werke, volume 8, p. 96-97), Gauss inverts the general elliptic integral of the first kind: he writes $\...
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Zeros of Jacobi elliptic functions

For the past month or so, I have been studying Jacobi elliptic functions $sn$, $cn$ and $dn$. However I stumbled upon a problem and it goes as following: I am trying to find $z \in \mathbb{C}$, such ...
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Closed form for the indefinite integral of the Jacobi Theta function

I am interested in the indefinite integral of $\vartheta_3(q;0)$. A lazy result gives us $$ \int\vartheta_3(q;0)\mathrm{d}q=q+2\sum_{n=1}^\infty \frac{q^{n^2+1}}{n^2+1}+C. $$ While there is nothing ...
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Proof of Jacobi fraction expansion in Triple Product Proof

In the proof of Jacobi's triple product identity by Jacobi, he considers the infinite product; $\frac{1}{(1-qz)(1-q^2z)...}$ and expands it into 1 + $\frac{B_1z}{(1-qz)}$ + $\frac{B_2z^2}{(1-qz)(1-q^...
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What are the general formulas for the transformations of the periods of the Weierstrass elliptic functions $\wp$?

On the first page of [Ritt 1922b], some functions that are related to the Weierstrass elliptic functions $\wp$ (see also) are listed. Which functions (c), (d) and (e) are these that can be read from ...
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Decomposition of a meromorphic function on the torus into the Weierstrass p-function and two automorphisms

Say $\tau $ be a complex number with $\Im(\tau)>0$.Then a meromorphic function $f$ on the torus $M$ can be thought of as a meromorphic function on the plane with $f(z+1)=f(z)$ and $f(z+\tau)=f(z)$ $...
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Elliptic Integral along complex path

I am interested in the following integral \begin{align} \int \frac{dt}{t\sqrt{(t-e_1)(t-e_2)(t-e_3)}} \end{align} where $e_1=\bar{e}_3$ and $e_2\in\mathbb{R}$ is negativ. Assume that the branch cut of ...
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Alternative series expansion for the elliptic integral of the second kind

I found the following alternative series expansion for the complete elliptic integral of the second kind (E): $ E(k) = \frac{(1+k')\pi}{4} \left\{ 1+\frac{1}{2^2}\left(\frac{1-k'}{1+k'}\right)^2+ \...
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Verification of Addition Theorem for the $\wp$-function

In Freitag's Complex Analysis, he briefly states the addition theorem for the Weierstrass $\wp$-function without giving a proof, specifying that "it is a simple computation and the details are ...
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Legendre's Complete Elliptic Integral of the 1st Kind - Calculating the argument

Given the definition of the complete elliptic integral of the 1st kind (see this link here), I am interested in finding a particular value of $k$ such that $$K(k) \equiv \int_0^1\dfrac{\operatorname{d}...
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Elliptic uniformization of $\sqrt{1+k+k^2}$

I have several elliptic polynomials $P_i(u)$ in Jacobi elliptic functions $sn(u|k)$, $cn(u|k)$ and $dn(u|k)$ with standard definitions $sn^2(u|k)+cn(u|k)^2=1$, $dn^2(u|k)+k^2sn^2(u|k)=1$. Coefficients ...
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Holomorphic function such that $\theta\left(z+\omega_j\right)=a_j\theta\left(z\right)$ satisfies $\theta\left(z\right)=ae^{bz}$.

I am trying to solve the following exercise: Let $\mathbb{Z}\omega_1+\mathbb{Z}\omega_2$ be a lattice in $\mathbb{C}$ and let $\theta$ be an entire function such that there exist $a_1,a_2\in\mathbb{C}...
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3 votes
1 answer
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Infinite sum of iterated integrals of matrix products

Edit: Discussion moved to Mathoverflow at https://mathoverflow.net/questions/395085/infinite-sum-of-iterated-integrals-of-matrix-products The problem: Let $$N(z) = \begin{pmatrix} 0 & \frac{1}{z} \...
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2 votes
1 answer
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What is a lattice of elliptic curve $y^2=x^3-x$?

It is well known that isomorphism class of elliptic curve and lattice up to homeothetic corresponds bijectively. But I don't have concrete examples. Can we figure out lattice from given weierstrass ...
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lattice and isomorphism class of elliptic curves over complex field is bijective?

Lattice and isomorphim class of elliptic curve corresponds bijectively? If we give lattice, we can define weierstrass form of elliptic curve, and gain one isomorphism class of elliptic curve. Does ...
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3 votes
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Derive singular value $\lambda(\sqrt{2}i)=(\sqrt{2}-1)^2$

Does anyone know how to prove that the following special value of the Modular Lambda Function is correct? $$\lambda(\sqrt{2}i)=(\sqrt{2}-1)^2$$ I have a somewhat promising observation that might help ...
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3 votes
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Integral over unit interval of the Nome:$\int_0^1q(x)dx=1-\int_0^1q^{-1}(x) dx$. No closed form necessary

After browsing for fun in Mathematics Stack Exchange trying to find an insight for a specialized infinite sine product, I came across the Nome which has to do with Q-Series and Elliptic Functions. I ...
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About the Laurent Series of the WeierstrassP function

Mathematica returns the following Laurent Series for the $\wp(z;(1,1+I))$ function Series[WeierstrassP[z, WeierstrassInvariants[{1, 1 + I}]], {z, 0, 12}] $$\frac{1}...
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1 answer
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$Φ$:$\Bbb C/\Lambda\to E(\Bbb C)$ : $t\mapsto (\wp(t),\wp'(t))$ is isomorphism as Rieman surface

If $\Lambda$ is a lattice in $\Bbb C$ the map $$z\mapsto (\wp(z),\wp'(z))$$ is a parametrisation of the complex points of the elliptic curve $$E:\qquad y^2=4x^3-g_2x-g_3$$ where $g_2$ and $g_3$ depend ...
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3 votes
1 answer
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When $\mathbb{C}/\Lambda$ is a Riemann surface?

Let $\Lambda$ be a lattice, that is $\Lambda=\{a\omega_1+b\omega_2\mid a,b\in\mathbb Z\}$. I heard that the necessarily and sufficient condition that $\mathbb{C}/\Lambda$ be a Riemann surface is $\...
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Jacobian elliptic function argument [closed]

I have a C++ code that computes jacobian elliptic sn, cn and ...
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Given a meromorphic function $f$ on $\mathbb{C}/\Lambda$, there is indeed a standard proof that $f \in \mathbb{C}(\wp,\wp')$

This question is from Silverman's 'the arithmetic of elliptic curves', p167. Given a meromorphic function $f$ on $\mathbb{C}/\Lambda$, there is indeed a standard proof that $f \in \mathbb{C}(\wp,\wp')$...
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