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

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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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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 ...
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100 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 ...
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1answer
31 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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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 ...
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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 ...
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1answer
22 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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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, ...
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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 ...
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1answer
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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 ...
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1answer
51 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: $$\omega'_2=a\omega_2+b\omega_1$$ ...
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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 $$\text{Im}(\tau)>0\;,\;-1<\text{Re}(\tau)\leq 1\;,\; |\tau-1/2|\geq1/2\;,\; |\tau+1/2|>1/2$$ ...
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1answer
32 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 ...
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Primitive of Weierstrass $\wp$

Consider a lattice $L=\mathbb{Z}+\mathbb{Z}\tau$. Take the function $\xi(z) = \frac{-1}{z} - \sum_{w \in L\backslash \{0\}} \Big ( \frac{1}{z-w} + \frac{1}{w} + \frac{z}{w^2} \Big )$. Obviously this ...
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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 + ...
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“Multiple angle” addition formulae for Jacobi elliptic functions

The addition formulae for the Jacobi elliptic functions are given by $sn(u+v)=\frac{sn(u)cn(v)dn(v)+cn(u)sn(v)dn(v)}{1-k^2sn^2(u)sn^2(v)}$, ...
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3answers
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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 ...
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1answer
59 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 ...
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182 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 ...
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149 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 ...
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36 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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103 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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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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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 ...
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2answers
48 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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24 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 ...
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1answer
47 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: $$ ...
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1answer
68 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 ...
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1answer
199 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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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 ...
4
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2answers
91 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 ...
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0answers
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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 ...
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1answer
68 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 ...
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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 ...
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1answer
46 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 ...
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1answer
57 views

Non linear system of differential equations

Is there a specific name to the following type of non linear ODEs $\begin{array}{c} \dot{x}_1 &= c_1 \, x_2\, x_3 \\ \dot{x}_2 &= \, c_2 x_1 x_3 \\ \dot{x}_3 &= c_3 \, x_2 x_1 ...
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Weierstrass-$\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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152 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,
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1answer
139 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 ...
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1answer
99 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 ...
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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 ...
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Elliptic function degree

If $f$ is an elliptic function (wrt a given lattice), and $R=p/q$ a rational function, then is there a formula allowing to compute the degree of the elliptic function $R(f)$ in Terms of the degree of ...
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1answer
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How can I conclude this “gluing property” for these Sobolev functions?

Let $u \in W^{1,2}(\Omega) \cap C(\Omega)$, where $\Omega$ is an open bounded domain in $R^n$ with smooth boundary.Let $B(x,R) \subset \overline{B(x,R)} \subset \Omega $ a ball. Consider $u^{\star} ...
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Group Operation of points on a Montgomery elliptic Curve (project coordinates)

I was trying to implement the double and addition formulas for elliptic points on a Montgomery elliptic curve. I came across this weird thing which should definitely not be happening. I took a point ...
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1answer
90 views

Elliptic curves and Weierstrass $\wp$ function - an example

Let $E: y^2 = 4x^3 - b$ an affine equation of an elliptic curve in $\mathbb{P}^2_{\mathbb{C}}$. Let $b$ be chosen such that the map $f: \mathbb{C} \rightarrow \mathbb{P}^2$ given by $z \mapsto ...
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1answer
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Double period except for poles

I'm trying to solve a problem in Complex Analysis whose function $f$ is defined in $\mathbb{C}$, is meromorphic and have double period $(f(z)=f(z+a)=f(z+b),\ \frac{a}{b} \notin \mathbb{R})$ except for ...
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1answer
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A problem about elliptic functions

I am trying to solve some problems in complex analysis, but I am not succeeding in the following problem. Suppose that $f$ is a function with the following properties: $f$ is non-constant; $f$ is ...
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1answer
38 views

Is this function defined in terms of elliptic $\mathrm{K}$ integrals even?

Let $R,z > 0$ be positive real constants, and consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by $$ f(v) = \frac{1}{\sqrt{(R+v)^2+z^2}}\ \mathrm{K}\!\left( \frac{4 R v}{(R+v)^2+z^2} ...
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1answer
35 views

The total number of poles of an elliptic function in $P_0$ is always $\geq$ 2

I'm trying to follow this proof from Stein & Shakarchi "Complex Analysis". The statement of the theorem is in the title of the question. The Proof is as follows: Suppose that $f$ (an elliptic ...
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1answer
30 views

Pole of elliptic function

Let $f:C→P1$ be such that $f(z+1)=f(z+i)=f(z)$ for all z∈C. Let $Γ=\{m+ni:m,n∈Z\}$. Show that if $f$ is holomorphic on $C∖Γ$, and $z⋅f(z)$ is bounded in a neighbourhood of $z=0$, then $f$ is ...