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Questions tagged [elliptic-functions]

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

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Addition formula for elliptic integral of second kind

Let $k\in(0,1)$ and the incomplete elliptic integral integral $E(u, k) $ be defined by $$E(u, k) =\int_{0}^{u}\operatorname {dn} ^2(t,k)\,dt\tag{1}$$ where $\operatorname {dn} (u, k) $ represents one ...
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1answer
35 views

Solving a 2nd order non-linear ODE

First off, please correct me if my title is wrong. I want to solve an equation which has the following form: $f'' + Af^3 + Bf = 0$ The closest I have gotten to such a form was when looking as the ...
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35 views

Jacobi sn: Does $w=\operatorname{sn}(z,m)$ send the line $\operatorname{Im}(z)=K'/2$ to the circle $\left \|w^4 \right \|=1/m?$

The picture below was generated in Mathematica and shows the image of a rectilinear grid in $\mathbb C$ under the elliptic mapping $\operatorname{sn}(z,m)$. The question here concerns the highlighted ...
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How to prove that a division of $\vartheta(z:\tau)$ functions has no poles

How do I prove that $f(z) + f(iz)$ has no poles? where $$f(z)=\dfrac{\vartheta^2(0;i)\vartheta^2(z+ \frac{1}{2};i)}{\vartheta^2(\frac{1}{2};i)\vartheta^2(z;i)}$$ and $$\vartheta(z;\tau)=\sum\limits_{n=...
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18 views

Jacobian elliptic function: unfamiliar definition

I am reading a physics book that uses a Jacobian elliptic function for a special purpose. For that i have to evaluate $\text{sn}(\zeta ; \gamma)$. However, both parameters are purely imaginary and the ...
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0answers
25 views

Weierstraß $\sigma$ function identity

Let $\Lambda$ be a lattice and $\sigma_{\Lambda_{\tau}}(z):= \sigma(z)= \prod_{w\in \Lambda\setminus \{0\}} \left(1 - \frac{z}{w}\right)\exp\left(\frac{z}{w}+\frac{z^2}{2w^2}\right) $ the Weierstraß ...
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Hyperelliptic function addition formula

$$x= \int_{0}^{f(x)} \frac{du}{\sqrt{1-u^2}\sqrt{1-k^2u^2}\sqrt{1-l^2u^2}}$$ $$f(0)=0$$ If we apply derivative operation for both sides, we get: $$f'(x)=\sqrt{(1-f^2(x))(1-k^2f^2(x))(1-l^2f^2(x))}$$...
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5answers
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$\sum_{\omega\in \mathbb{Z}(i)^*} |\omega|^{-2}$ does not converge.

I'm trying to prove that the series $$ \sum_{\omega\in \mathbb{Z}(i)^*} |\omega|^{-2} $$ The problem can be viewed as the sum above the fundamental region $\Omega^* = \{m\omega_1 + n\omega_2 : m,n\...
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Is it true that $\int_0^1 \big(K(k^{1/2})\big)^2\,dk = \frac{7}2\zeta(3)$?

Define the complete elliptic integral of the first kind as, $$K(k) = \tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)$$ Part I. From the link above, we find some of the evaluations below, ...
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What is $\int_0^1 \left(\tfrac{\pi}2\,_2F_1\big(\tfrac13,\tfrac23,1,\,k^2\big)\right)^3 dk$?

As in this post, define the ff: $$K(k)=K_2(k)={\tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)}$$ $$K_3(k)={\tfrac{\pi}{2}\,_2F_1\left(\tfrac13,\tfrac23,1,\,k^2\right)}$$ $$K_4(k)={\...
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46 views

Nonlinear ODE - possible transformations or analytical techniques?

I work with a problem that gives me a nonlinear 3rd order ODE. The equation in question is: $f'''(x) - \alpha f(x) f''(x) + \alpha f'(x)^2 + k f'(x)=0$ The coefficient $k$ always turns out to be an ...
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Is this an elliptic curve $x^2+y^2=k^2(1+x^2y^2)$

Is this an elliptic curve $x^2+y^2=k^2(1+x^2 y^2)$ over rationals? If yes, how can we transform it into usual form? My friend sent me this curve.
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1answer
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An intriguing pattern in Ramanujan's theory of elliptic functions that stops?

I. Define the ff integrals, $$K(k)=K_2(k)=\int_0^{\pi/2}\frac{1}{\sqrt{1-k^2 \sin^2 x}}dx=\large{\tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)}$$ $$K_3(k)=\int_0^{\pi/2}\frac{\cos\left(...
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Zeroes of some degree of two elliptic functions

Let $\tau \in {\mathbb C}$ with $\mathrm{Im} \tau > 0$, $a,b \in {\mathbb Q}$ not both integers (it's not clear to me whether assuming only $a,b \in {\mathbb R}$ will make a difference to the ...
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0answers
66 views

About Sturm's bound

The next theorem is known as Sturm's bound. Theorem:Let $\mathfrak{m}$ be a prime ideal in the ring of integers $\mathcal{O}$ of a number field $K$, and let $\Gamma$ be a congruence subgroup of of ...
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Finding a particular solution to a linear PDE

I want to solve the PDE $$\frac{\partial u}{\partial t}+x_1(x_2-x_3) \frac{\partial u}{\partial x_1}+x_2(x_3-x_1) \frac{\partial u}{\partial x_2}+x_3(x_1-x_2) \frac{\partial u}{\partial x_3}=\sum_{i=1}...
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1answer
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Showing that an elliptic function has no poles

Let $\Lambda = \{m \omega_1+n\omega_2; m,n \in \mathbb{Z}\}$ with $\omega_i \in \mathbb{C}$ with $\omega_2/\omega_1 \notin \mathbb{R}$ be a lattice. Define the Weierstrass $\mathscr{P}$ function on ...
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1answer
37 views

elliptic curve over GF(2^3)

This is $f(x) = x^3+x+3$ over $GF\left(2^3\right)$ How to know numbers of points on this equation? How to find those points? Is it an irreducible polynomial?
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1answer
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Evaluating a derivative in gp/pari

Specifically, I'm trying to evaluate the derivative of the Weierstrass P function at a specific point. I know that I can set up a function like the following p(z) = ellwp([1,I],z); Which will ...
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1answer
52 views

How to show that Jacobi sine function is doubly periodic

The Jacobi sine function can be defined using two definitions. The first is $\operatorname{sn}(u,m)=\sin(\phi)$ where $u=\int_0^{\phi}\frac{d\theta}{\sqrt{1-m\sin^2\theta}}$. Alternativey it may be ...
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1answer
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Hektic Oscillator

It's fascinating that whereas the solution of the quartic oscillator (restoring force $\propto r^3$) problem can be expressed as the distance-from-origin of a point moving with unit speed along a ...
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1answer
40 views

Calculation of Elliptic Functions at Extreme Values of Eccentricity

The scenario of transverse oscillations of a mass in centre of an elastic string yields an oscillator with a restoring force that is purely cubic ... or a potential well that is quartic. The solution ...
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69 views

Questions related to the Riemann Xi function $\xi(s)$ and Jacobi theta functions $\vartheta_3(0,q)$

This question assumes the following definitions. (1) $\quad\psi(x)=\sum\limits_{n=1}^\infty e^{-\pi\,n^2\,x}=\frac{1}{2} \left(\vartheta_3\left(0,e^{-\pi\,x}\right)-1\right)$ (2) $\quad f(x)=\sum\...
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0answers
49 views

Addition formula for $\text{sn}(u)=\text{sn}(u,k)$

I have just learned the definition of the first Jacobian elliptic function $\text{sn}(u)=\text{sn}(u,k)$, defined as the sine of the inverse function of $$F(\phi,k):=\int_0^\phi \frac{dx}{\sqrt{1-k^2\...
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Location of the zeros of Dedekind Eta Function

Just a fast question, since I have not been able to find any answer for it online. Where are the zeros of Dedekind eta function $\eta(s)$ located? Apart from the trivial one as $s \to i \infty$, ...
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0answers
128 views

Show that by a translation poles on the boundary of an elliptic curve can be avoided.

In Complex Analysis by Freitag it is claimed that if there are poles on the boundary of an elliptic curve (the parallelogram) can be avoided by a-translation. Is there any simple but rigorous proof ...
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0answers
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Finding doubly periodic solutions to partial differential equations

Say I have a certain PDE in real variables $x$ and $y,$ which might be nonlinear, so that we can't necessarily just throw a Fourier series at it. By way of some intuition, let's say, I have a very ...
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1answer
33 views

Is there a function with prescribed zeros and poles on an elliptic curve?

Let $T$ be the complex tore from the lattice $(1, \tau)$ where $im(\tau)>0$. How to prove the existence of a meromorphic function on $T$, with divisor $(0) + (\frac{1}{2}) - 2 (\frac{\tau}{2})$ ? (...
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0answers
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Exact value of Elliptic Integrals.

I was taking currently in a elementary calculus course where i found how to find arc lengths of a smooth continuous curve. so here is how i started : $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\Rightarrow y=...
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1answer
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Quotient of quarter periods K' and K of Jacobi elliptic functions

There are several ways to express the quarter period $K$, $$ K(m)=\int_0^{\pi/2}\frac{\mathrm{d}\theta}{\sqrt{1-m\sin^2\theta}}, $$ as a power series (and thus for $K'=K(1-m)$ there are, too) and also ...
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Problem with the $\begin{eqnarray*} \wp(z) \end{eqnarray*}$ differential equation of Weierstrass

The Laurent series of $\wp(z)$ about $z=0$ is given by $$ \wp(z) = z^{-2}+\sum_{k=1}^\infty (2k+1)G_{2k+2}z^{2k} $$ and for $z\in\mathbb C$ and $z\notin \Lambda$ (=lattice) we have $$ (\wp(z)')^...
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50 views

Reference Request - Elliptic Functions

From what I hear, elliptic functions were orginally introduced to find roots of polynomials, especially to help compute integrals. What is a book that emphasizes this motivation and approach to ...
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62 views

Correspondence between a complex torus and an elliptic curve

In The Arithmetic of elliptic Curves, on page 156, Silverman considers the map $$\phi\colon\mathbb C/\Lambda\longrightarrow E\subset \mathbb P^2(\mathbb C)$$ $$z\longmapsto [\wp(z),\wp'(z),1]$$ where $...
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0answers
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Unable to Understand the Abel's Theorem for Elliptic Functions

Definition: Let $f$ be an elliptic function on $\mathbb{C}$ and $\Omega \subset \mathbb{C}$ be the periodic lattice of $f$. The divisior of $f$ is defined as $$(f)=\sum_j n_jP_j$$ where $P_j$ are the ...
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0answers
71 views

Evaluation of complicated multiple integral

I have the following multiple integral I would wish to evaluate $$\int_0^{2\pi} dx_1 dx_2 \cdots dx_n D(x_1,x_2)\cdots D(x_n,x_1)$$ where the $D$ functions are given by $$ D(x,x') = \text{Sn}^2\...
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Explicit Values of the Jacobi Theta Function [duplicate]

The sum $\sum_{n=-\infty}^{\infty}\exp(-\pi n^2) $, or $\vartheta(0;i)$ (Jacobi Theta Function) has a closed form solution of $\frac{\pi^{\frac{1}{4}}}{\Gamma(\frac{3}{4})}$ but nowhere have I been ...
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1answer
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$\wp(z)$ and points of order $3$

This problem comes from Koblitz's book Introduction to Elliptic Curves and Modular Forms page 41 problem 2 and it says Let $$f_N(z) = N \Pi(\wp(z)-\wp(u))$$ Where the product is taken over nonzero $...
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1answer
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The solutions of $(y')^2=P(y)$ with $\deg P \in \{3,4\}$.

Here is some background on my question. The Weierstraß function (as well as its translates) $\wp(x)$ solves the implicit first-order ODE $$(y')^2=4y^3-g_2y-g_3.$$ Differentiating gives the second-...
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1answer
83 views

Converting a function of $\cos^2$ to a complete elliptic integral of the first kind

I am having a hard time following Equation 2.6 of Taib, Bachok Bin. "Boundary integral method applied to cavitation bubble dynamics." (1985). The equation is on the middle of page 8 of the document or ...
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4answers
267 views

How to derive relationship between Dedekind's $\eta$ function and $\Gamma(\frac{1}{4})$

I am trying to determine in what way to approach finding a connection between Dedekind's Eta Function, defined as $$\eta(\tau)=q^\frac{1}{24}\prod_{n=1}^\infty(1-q^n)$$ where $q=e^{2\pi i \tau}$ is ...
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1answer
35 views

$| \Re(\tau)| \leq 1/2$ for $\tau$ in fundamental region.

I've been working through Alfors section on elliptic functions and I'm stuck on an inequality in one of the proofs regarding the fundamental region (thereom 2 of 2.3 The canonical basis). We have the ...
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1answer
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Properties of $\frac {\theta_{1}''(z|\tau)}{\theta_{1}(z|\tau)}$?

I'm looking for references concerning the properties of the function $\frac {\theta_{1}''(z|\tau)}{\theta_{1}(z|\tau)}$ where $\theta_{1}(z|\tau)$ is a Jacobi theta function defined here. I am trying ...
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1answer
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Find extreme points of a rotated ellipse equation on a given axis

I'm having hard time figuring out how to find the points where it is most extreme on the X and Y axis. For example lets say I have an equation that describes an ellipse that is rotated: (x * RadiusX ...
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1answer
35 views

Determine points on an eliptic curve when p is large? + Point at infinity clarification

I have come across this question here: Points on elliptic curve over finite field Which describes how you can take every value $x$ from $0$ to $p-1$ and calculate the points of an elliptic curve. ...
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1answer
60 views

Calculating the lattice of the tori of a non-singular projective cubic curve

If $C$ is the curve in $\mathbb{C}\mathbb{P}^{2}$ defined by the zero set of the polynomial $P^{\lambda}(x,y,z) = y^{2}z - x(x-z)(x-\lambda z)$, for $\lambda$ not $0$ or $1$. Then we know that $C$ is ...
4
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2answers
215 views

How to show that cusp of congruence subgroup $\Gamma_0(4)$ is $0,\frac{1}{2},\infty$?

Let $\Gamma_0(4)$ be a congruence subgroup of $SL(2,\mathbb{Z})$ defined as $$\Gamma_0(4)=\Big\{M=\begin{pmatrix} a &b\\ c& d \end{pmatrix}\in SL(2,\mathbb{Z}) | c \equiv 0\bmod 4\Big\}.$$ ...
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1answer
50 views

Triangle Inequality: use to prove convergence (psi function elliptic functions)

The problem statement, all variables and given/known data Attached I understand the first bound but not the second. I am fine with the rest of the derivation that follows after these bounds, 2. ...
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2answers
49 views

How can I draw a function of the nome $q$ on the unit disk?

This figure is the real part of the discriminant as a function of the nome $q$ on the unit disk. It is taken from the wiki link: https://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions I ...
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2answers
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Finding zeroes and poles of the Weierstrass $\wp$ and $\wp^{'}$ function associated with a lattice.

I have a lattice, $\Omega$, with basis $\lbrace 2+i, 1+3i\rbrace$ and fundamental region (square) $P$ with vertices $1+2i,\ 2, \ -1+i\ $ and $-i$. I want to find the zeroes and poles of $\wp$ and $\wp^...
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0answers
58 views

Higher derivatives $\wp^{(2n)}$ of the Weierstrass $\wp$-function

I'm interested in considering the Weierstrass $\wp$-function denoted $\wp(\tau, z)$ as well as its higher derivatives $\wp^{(2n)}$, for all $n \geq 0$. The derivatives are with respect to the $z$ ...