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

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6
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1answer
55 views

Curious integrals for Jacobi Theta Functions $\int_0^1 \vartheta_n(0,q)dq$

There are various identities for the Jacobi Theta Functions $\vartheta_n(z,q)$ on the MathWorld page and on the Wikipedia page. But I found no integral identities for these functions. Meanwhile, ...
0
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0answers
6 views

jacobi_sn with theta functions and complex argument, test in C++ and wxMaxima

I didn't know whether to ask this in S.O. or here, so if it's the wrong place, I'm sorry. I need an implementation of jacobi_sn in C++ and I chose the equivalent theta functions version (...
0
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0answers
32 views

Elliptic curve, different forms of.

$y^2 = x^3 + mx + c$ An elliptic curve in the form defined in Wikipedia $y^2 = x(x-A)(x+B) = x^3 +(B-A)x^2 + ABx$ Frey's curve has no term in $x^2$, but $2$. does because from Fermat, $A=a^n$ ...
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15 views

How to show that this Jacobian elliptic function always has two real zeroes?

I have the following elliptic function $f(x)$: $$f(x) = -\ dn^{2}x \ cn^{2}x \left(3m \ sn^{2} x + \frac{E'}{K'}\right) + \ sn^{2} x \left(m \ cn^{2} x + \ dn^{2} x \right) \left(m \ sn^{2} x + \frac{...
0
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0answers
25 views

How to simplify the following expression involving Jacobian elliptic functions?

I would like to show that a certain elliptic function $F(x)$ (that is periodic, say with some period $h$) has exactly two zeroes in $[0,h)$. Let us recall some notation. Given a parameter $m \in [0,1]$...
3
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0answers
86 views

Equivalent of Jacobi elliptic functions

I would like to compute the equivalent function when $k\rightarrow 0$ of $$\frac{1}{k}\,\left(\,Z(\frac{u}{k},k) - dn(\frac{u}{k},k)\,\right)$$ where $k$ is the modulus of the Jacobi elliptic ...
0
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25 views

Elliptic functions

I would like some references to compute such limits: Jacobi elliptic functions: $\lim_{k\to 0} dn(z/k,k)$ Second species: $\lim_{k\to 0} Z(z/k,k)$ I didn't find what I wanted in the Lawden's book. ...
2
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0answers
29 views

Jacobi modulus and Weierstrass $\wp$

Let $0 < k < 1$ and $$K := \int_0^{\pi/2} \frac{1}{\sqrt{1 - k^2 \sin^2(\theta)}} \, \mathrm{d}\theta, \; \; K' := \int_0^{\pi/2} \frac{1}{\sqrt{1 - (1-k^2) \sin^2(\theta)}} \, \mathrm{d}\theta.$...
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12 views

Can someone explain the Hermite quintic solution in more detial

From https://en.wikipedia.org/wiki/Bring_radical How does he obtain this quartic? and calculate the required elliptic modulus $k$ by solving the quartic equation: $k^4 + A^2k^3 + 2k^2 - A^2k + 1 = ...
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14 views

“Opposite” point on ellipsis by axis (or vector)

I'm currently working on a little game and am stumped as to how I'd solve this math problem. What I'm trying to do is get the "rotation" needed for B, where B is always opposing A on the Y axis no ...
0
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2answers
38 views

Laurent Series Expansion Logic

Relative to the below image, I am curious about the progression from equation 3.2 to equation 3.3, then from equation 3.3 to equation 3.4. I understand the logic in 3.2. I understand that a Laurent ...
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0answers
10 views

Calculating X & Y coordinates of a point that is perpendicular to an ellipse point AND offset by -5

I am trying to calculate an offset point from a point on an ellipse - I need to be perpendicular to each point on the ellipse but 5 points in from the point on the ellipse. The result will probably ...
4
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145 views

Algebraic equation concerning the Jacobi sn function

What are the solutions $u$ to the equation $\sqrt{1-m} \operatorname{sn} (u \mid \frac{1-m}{1+m}) = 1$ for a given $m$? Because $\mbox{sn}$ is an elliptic function of order 2, we know it attains any ...
0
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1answer
39 views

How to calculate an inner ellipse points that is always a set distance from an outer ellipse points

I have an Ellipse with known coordinates , I would like to know how I can create an inner ellipse coordinates that are exactly 5 inches perpendicular from the outer ellipse points. Please see the ...
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8 views

If $f\in H(\mathbb{C}\setminus \Gamma)\cap D.P.$ and $|zf||_{|z|=\varepsilon}<\infty$, then f=constant

If $f\in H(\mathbb{C}\setminus \{m+ni:m,n\in \mathbb{Z}\})$, doubly periodic with periods 1,i and $zf$ bounded in nbhd of origin, then f=constant. Any mistakes By periodicity $\infty>z\cdot f(z)=...
1
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0answers
13 views

Expressing Elliptic functions as ratio of $\wp$ and $\wp'$

I was reading Stein's book on Complex analysis, and on page 271 they say that every elliptic function $f$ with periods 1, and $\tau$ is a ratio of $\wp$ and $\wp'$. The book doesn't have a proof of ...
1
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1answer
25 views

How can I identify whether a point is within an ellipse which is not orthogonal in orientation

I'm looking for an equation which will tell me whether or not a point in two-dimensional space, is located within an ellipse of known dimensions and orientation, and that is not orthogonal in nature.
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17 views

Translating from (short) Weierstrass form to a Lattice. [duplicate]

I know that given an elliptic curve in the form $\mathbb{C}/L$ for L some lattice, $\mathbb{Z}+\mathbb{Z}\tau$, then we can use the Weierstrass $\wp$ functions to turn it into short Weierstrass form, $...
1
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1answer
27 views

Trying to derive the Laurent Expansion of $\wp$ (Weierstrass Elliptic function)

I have the definition of the Weierstrass elliptical function $\wp$ as: $$\wp(z)=\frac{1}{z^2}+\sum_{\omega\neq 0}\frac{1}{(z-\omega)^2}-\frac{1}{\omega^2}$$ Where $w=n\omega_1+m\omega_2$, for $\...
5
votes
1answer
101 views

May I know how this integral was evaluated by using the theory of elliptic integrals?

I can not solve the following integral using the theory of elliptic integrals: $$\int_a^b \frac{\sin(x)}{\sqrt{c-\sin(x)}}dx$$ Where $a\geq 0, b>0, c>0$. Wolfram$|$Alpha showed the following ...
0
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1answer
54 views

Functional equations of $\lambda(\tau)$

We have the elliptic lambda function: $$\lambda(\tau)=\frac{e_3-e_2}{e_1-e_2}$$ We want to look at how $\lambda$ changes under a modular transformation: $$\omega'_2=a\omega_2+b\omega_1$$ $$\omega'_1=c\...
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34 views

Problem understanding elliptic lambda function $\lambda(\tau)$ fundamental region in Ahlfors text.

With the modular lambda function I understand the fundamental region to be the region bounded by $$\text{Im}(\tau)>0\;,\;-1<\text{Re}(\tau)\leq 1\;,\; |\tau-1/2|\geq1/2\;,\; |\tau+1/2|>1/2$$ ...
2
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1answer
34 views

Problem completing proof that the Weierstrass $\wp$ function is uniformly convergent

I'm having trouble following Ahlfors logic in his text concerning the proof that the $\wp$ function is convergent. The proof comes down to whether the series $$\sum_{\omega\neq 0}\frac{1}{|\omega|^3}...
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0answers
60 views

Property of Weierstrass sigma function

In theorem 1.2.3 of Schertz' Complex Multiplication says that For any $\omega \in \mathcal{L}$, a fixed lattice, we have the property: $$ \sigma(z + \omega) = \psi(\omega)e^{\eta(\omega)(z + \omega/...
3
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0answers
31 views

“Multiple angle” addition formulae for Jacobi elliptic functions

The addition formulae for the Jacobi elliptic functions are given by $sn(u+v)=\frac{sn(u)cn(v)dn(v)+cn(u)sn(v)dn(v)}{1-k^2sn^2(u)sn^2(v)}$, $cn(u+v)=\frac{cn(u)cn(v)-sn(u)dn(u)sn(v)dn(v)}{1-k^2sn^2(...
1
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3answers
115 views

Elliptic Integrals

In my homework I had to solve the following integral $\displaystyle\int_0^\pi \mathrm{d}\Psi \frac{\cos\Psi}{\sqrt{1+2s(1-\cos\Psi)}}$ with some constant $s\ll1$ The solution said this is an "...
2
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1answer
69 views

The derivation of the Weierstrass elliptic function

I am wondering if any of you could point me to any books and/or lecture notes that explain the Weierstrass $\wp$ function for a self-studying student of elliptic curves and functions. I am interested ...
10
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1answer
206 views

Elliptic functions as inverses of Elliptic integrals

Let us begin with some (standard, I think) definitions. Def: An elliptic function is a doubly periodic meromorphic function on $\mathbb{C}$. Def: An elliptic integral is an integral of the form $$f(...
7
votes
1answer
152 views

The elliptic integral $\frac{K'}{K}=\sqrt{2}-1$ is known in closed form?

Has anybody computed in closed form the elliptic integral of the first kind $K(k)$ when $\frac{K'}{K}=\sqrt{2}-1$? I tried to search the literature, but nothing has turned up. This page http://...
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0answers
39 views

Zeta function of $y^2 = x^3 - x$ over Fp

Zeta function of $y^2 = x^3 - x$ over Fp, where p = 3(mod 4) Can someone give an explanation of a zeta function? I've tried researching it, and I cannot seem to understand. Is there some kind of ...
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0answers
105 views

Differentiating a period of an elliptic curve under the integral sign

Let $$g = \frac{27J}{1 - J},$$ where $J$ is the absolute invariant, and define $$\Omega = \int_{\gamma(J)} \frac{dz}{\sqrt{4z^3 + g(z + 1)}}.$$ Here, $\gamma(J)$ is a contour in the complex plane that ...
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22 views

Need clarity on calculating the y coordinate in elliptic curve cryptography

I'm just new to elliptic curve cryptography. I have been working on RSA for quite some time. Moreover I'm not from a mathematical background. The whole concept looks very complex. So tell me my ...
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0answers
7 views

Why are the points of a period module necessarily isolated?

Given a non-constant meromorphic function $f(z)$ then how can we say that the absence of a finite accumulation point of our module $M$ implies that $f$ is constant? I understand that an accumulation ...
1
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2answers
49 views

How does this differential equation define an oscillation from a to b?

The differential equation reads: $$ \dot{y}^2+(y^2-a^2)(y^2-b^2)=0 $$ where $a,b\in \mathcal{R}$ and $a<b$. It is claimed that this differential equation defines an oscillation from $a$ to $b$. ...
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26 views

advice for curve fitting

I have numerically obtained some curves, corresponding with it I have also obtained some roots. I strongly believed these curves can be fitted with some (elliptic) functions taken the roots as ...
0
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1answer
53 views

Using Jacobi Elliptic Functions to solve Euler's Equations of Motion

Recently, I have been trying to use Jacobian Elliptic Functions to solve for angular velocities in the Euler's Equations of Motion, which look like this: $$ I_1\dot\omega_1-\omega_2\omega_3(I_2-I_3)=0\...
1
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1answer
72 views

Inverse Elliptic function question

Let $\psi$ be an elliptic function with periodic lattice $\mathbb{Z}[\omega]= \mathbb{Z}\omega \oplus \mathbb{Z}$, a pole of order $2$ at $0$, and simple zeros at $\pm\dfrac{\omega-1}{3}$. Here $\...
4
votes
1answer
224 views

How to solve the Brioschi quintic in terms of elliptic functions?

Given the Brioschi quintic $$w^{5}-10cw^{3}+45c^{2}w-c^2=0$$ I'm interested in seeing different ways of solving it in terms of elliptic functions or theta functions.
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79 views

conjectured generalization of euler's formula.

Given the elliptic modulus $k$ ,such that the complementary modulus is defined by $k'\equiv \sqrt{1-k^2}$,the jacobi amplitude $\phi\equiv am(u|k)$ and $K(k)$,is the complete elliptic integral of the ...
4
votes
2answers
95 views

Does the elliptic function $\operatorname{cn}\left(\frac{2}{3}K\left(\frac{1}{2}\right)\big|\frac{1}{2}\right)$ have a closed form?

Given the complete elliptic integral of the first kind $K(k)$ for the modulus $k$, can the elliptic function $$\text{cn}\left(\frac{2}{3}K\left(\frac{1}{2}\right)\bigg|\frac{1}{2}\right)$$ be ...
2
votes
0answers
48 views

Expansion of Weierstrass elliptic function in second period

I would very much like to find expansions of the Weierstrass $\wp$ and $\zeta$-functions for small absolute values of the second period $\omega_2$. So, more precisely, I would like, for $|\omega_1|\...
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1answer
80 views

Show that Weierstrass function is elliptic function.

Prove that Weierstrass function is periodic with respect to lattice $L (L\subset \mathbb{C})$ .i-e $f(z+w,L)=f(z,L)$ ($w\in L$). $f(z,L)=\frac{1}{z^2}+\sum_{0\ne w\in L}\Big(\frac{1}{(z-w)^2}-\frac{1}...
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0answers
56 views

Elliptic formulation of sum with cosine and hyperbolic cosine

Can anyone check my reformulation below?$$\sum_{k\in \mathbb{Z}}\frac{1}{\cosh{(k+z)\pi}+\cos{(k+z)\pi}}=\frac 2\pi \csc{\pi z}D_z\left[\tan^{-1}\left(\frac{\theta(z/2,2)}{\vartheta(z/2,2)}\right)\...
3
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1answer
49 views

Group operations on Montgomery Curves in affine representation

I'm trying to understand group operations on elliptic Montgomery curves in affine representation. Let's say the curve I use is Curve25519, i.e.: $$y^2 = x^3 + A\,x^2 + x\quad\text{where}\quad A=...
2
votes
1answer
60 views

Non linear system of differential equations

Is there a specific name to the following type of non linear ODEs $\begin{array}{c} \dot{x}_1 &= c_1 \, x_2\, x_3 \\ \dot{x}_2 &= \, c_2 x_1 x_3 \\ \dot{x}_3 &= c_3 \, x_2 x_1 \...
3
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0answers
39 views

Weierstrass-$\wp$ Function Asymptotics

Given the Weierstrass-$\wp$ function, $$\wp(2x+1+\tau \mid 1, \tau),$$ with half-periods $1$ and $\tau=\omega_2/ \omega_1$, I want to look at the case where $\rm{Re}(\tau) \in \mathbb{Z}$ and I want ...
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2answers
171 views

map the UHP to an equilateral triangle

Explain how the upper half-plane can be mapped one-to-one and conformally onto an equilateral triangle. Thanks,
6
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1answer
153 views

Animation of Weierstrass $\wp$-function as a map from a torus to the sphere?

I am wondering if there exists somewhere an "animation" of one such map (for some lattice / torus), in the style of the kind of $z \mapsto z^2$ maps one encounters in complex analysis classes (one can ...
4
votes
1answer
110 views

Jacobi Elliptic Functions built from Jacobi theta functions

I believe I understand the general theory of elliptic functions to an extent. What I can't seem to find is the distinct method which was used to show that a particular combination of Jacobi Theta ...
0
votes
2answers
51 views

How to show an ODE system has no global solution

Starting from any $(x_0,y_0,z_0)\in \mathbb{C}^3$, can the following ODE system have a solution for all real number? \begin{align} x'(t) &=3 y^2(t) \\ y'(t) &=2 x(t) z(t)-1 \\ z'(t) &=0 ...