Questions on doubly periodic functions, like Jacobi and Weierstrass elliptic functions.

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61 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
25 views

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
51 views

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
24 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
17 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
23 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 ...
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112 views

Elliptic Function

Let $y$ be the function defined by $$y(\theta)=2sin\frac{\theta}{2}\prod_{k=1}^{\infty}\frac{(1-e^{i\theta}q^k)(1-e^{-i\theta}q^k)}{(1-q^k)^{2}}$$ where $q = e^{2\pi i\tau}$ Show that $y$ has simple ...
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41 views

Elliptic Curve Group and Multiplicative Inverse of an element.

Suppose $E$ be an Elliptic Curve over a field $F_q$ and $q=p^n$ where $p=$ prime. We know that the Elliptic Curve group $E(F_q)$ under addition is an Abelian/Commutative Group of order, ...
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convert JacobiSN to JacobiDN identity

I have a Jacobi Elliptic sn function: \begin{equation} 0.9258200998{\it sn} \left( 0.7559289460\,It-{\it ns} \left( - 1.154700538,- 0.6123724359\,\sqrt {2} \right) ,- 0.6123724359\,\sqrt {2} ...
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1answer
100 views

Integral relations in Fricke and Klein

Can someone please explain how Fricke and Klein obtain the integral relationa stated at the top of p. 34 in this book? The entire book can be previewed on Google Books. It is an old book and I do not ...
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255 views

The equation $x^2 + y^2/b^2 = 1$ [closed]

The page "Jacobi elliptic functions" on Wikipedia has the heading "Definition by inverses of elliptic integrals". But where do the elliptic integrals come from? There is no need to mention the arc ...
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21 views

If the result of differentiating a function converges can we claim that there are no singularities in the function?

I was trying to understand an answer to another question of mine Showing Weierstrass Elliptic Function is meromorphic in which the answerer has used "You can differentiate the function term by ...
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1answer
61 views

Showing Weierstrass Elliptic Function is meromorphic

Consider the Weierstrass Elliptic function, $$\rho(z) = \frac{1}{z^{2}} + \sum\bigg(\frac{1}{(z-m-nw)^{2}} - \frac{1}{(m+nw)^{2}}\bigg), $$ where $m,n \neq 0,0 $. How can one show that it is ...
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37 views

Proving a function elliptic.

Consider the elliptic function, $\rho(z) = \frac{1}{z^{2}} + \sum(\frac{1}{(z-m-nw)^{2}} - \frac{1}{(m+nw)^{2}}) $ here $m,n \neq 0,0 $. The task is to prove this function elliptic, so doubly periodic ...
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An Elliptic function can not be holomorphic/analytic?

I was reading about elliptic functions on the wiki and it said that a doubly periodic meromorphic function in contention of being an elliptic function can not be analytic/holomorphic as it would then ...
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43 views

Generality of a solution of a nonlinear PDE

Given the nonlinear PDE $$ \partial^2\phi+\phi^3=0 $$ I consider the characteristics $\xi=\kappa\cdot x$. Then I look for a solution in the form $$ \phi(x)=a\cdot\chi(\xi). $$ Provided ...
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$\exists C \in \mathbb{C}^{\times}$, $n\in\mathbb{N}$: $f = C\Delta^n$

Let $f \in M_k(\Gamma)$ not null in $\mathbb{H}$. I want to show that there exists a $C \in \mathbb{C}^{\times}$ and $n\in\mathbb{N}$ with $f = C\Delta^n$. I think one can show that for a $k>0$ ...
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1answer
35 views

Boundedness of a modular form in $\mathbb{H}$

Let be $k>0$ and $f \in S_k(\Gamma)$. I want to show that the function $h(z)=Im(z)^{\frac{k}{2}}\cdot |f(z)|, \; z\in\mathbb{H}$ is bounded in $\mathbb{H}$. I have already shown, that $h$ is ...
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84 views

Inverse of elliptic integral of second kind

The Wikipedia articles on elliptic integral and elliptic functions state that “elliptic functions were discovered as inverse functions of elliptic integrals.” Some elliptic functions have names and ...
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Graphing elliptical curves based on group operation

I just found this and it blew my mind (he gives an elliptical curve to do multiplication). If I understand correctly (from reading the link and other things) the Abelian group he is using is ...
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56 views

When does $\wp$ take real values?

Whittaker and Watson mention that when the invariants of Weierstrass $\wp$ function are such that $g_2^3 - 27g_3^2 > 0$, and if $2\omega_1$ and $2\omega_2$ are its periods then the function takes ...
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1answer
32 views

About $\wp$ function with certain conditions

Let $\wp$ denote the Weierstrass elliptic function and $\wp^{'}$ its derivative. Now, consider a two dimensional lattice $\Lambda\subset\mathbb{C}$ and let $\Lambda_{2}=\lbrace ...
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57 views

Elliptic function $f(z)=\frac{a_{-2}}{z^2}+a_0+a_1z+a_2z^2+\dots$ must be even

Let $f:\mathbb{C}\longrightarrow \mathbb {C}$ be a nonconstant elliptic function such that $$f(z)=\frac{a_{-2}}{z^2}+a_0+a_1z+a_2z^2+\dots$$ How to prove that $f(z)$ is even. Notice that ...
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1answer
27 views

Extremizing a functional with a non-elementary solution to the Euler-Lagrange equation

Someone recently asked this question, but deleted it before I could post an answer. I figured I might as well share it for future internet travellers. The question is problem $10$ here. It states ...
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1answer
32 views

Elliptic functions $f(z+\lambda_1)=af(z) \; , \; f(z+\lambda_2)=bf(z) $

Let $\lambda_1$ and $\lambda_2$ be complex numbers with nonreal ratio. Let $f(z)$ be an entire function and assume there are constants $a$ and $b$ such that $$f(z+\lambda_1)=af(z) \;\;\;\;,\;\;\;\; ...
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18 views

Give me some examples of elliptic operators and Fredholm operators or explain its difference

First, I know Fredholm operator are kinds of elliptic operator which satisfies its kernels and cokernels have finite-dimension. And in generally, elliptic operators on a compact manifold are fredholm ...
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1answer
25 views

Harmonic function on an open set is infinite dimensional

I want to prove that the space of harmonic function on an open set $\Omega\subset R^N$ $(N\geq 2)$ is uncountablely infinite dimensional. That is, I want to prove that $$A:=\{u\in ...
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44 views

Sources on Jacobi Elliptic Functions

I'm interested in learning more about the Jacobi Elliptic Functions and the associated theta functions. For instance, what was the initial motivation for defining them? What are some applications? Is ...
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28 views

$\wp$ via Jacobi triple product

$$\wp(z;\tau) = -(\log \vartheta_{11}(z;\tau))'' + c $$ $$\vartheta_{11}(z|q) = -2 q^{1/4}\sin(\pi z)\prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 - 2 \cos(2 \pi z)q^{2m}+q^{4m}\right)$$ Then ...
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1answer
65 views

Modular property of Weierstrass $\wp$ function

For $\tau\in \mathbb H=\{ x+iy\in \mathbb C \lvert x,y\in \mathbb R, \, y>0\}$ and $z\in \mathbb C$, let us define $$ \wp=\wp(\cdot,\tau): \mathbb C \rightarrow \mathbb P^1\, , \quad z\mapsto ...
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Recursive identity for elliptic lattice constants $\sum_{\lambda\in\Lambda\setminus0} \lambda^{-2k}$

I am stuck on Exercise 3 in these notes. To keep this question self-contained: we have $\displaystyle\Lambda=\langle\omega_1,\omega_2\rangle=\omega_1\Bbb Z+\omega_2\Bbb Z\subset\Bbb C,$ ...
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Finding the inverse of an integral

I'm looking for a computational approach here, since I don't think there is a closed-form solution. I have the following: $$ s(x) = \rho + \int_{\rho}^{x} \sqrt{ 1 + (\alpha \cos t - k)^2 } \, dt $$ ...
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91 views

Quantities $g_2$, $g_3$, $\Delta$

This question is somewhat related to this one. Let $\lambda$ be the modular lambda function. Greenhill (Elliptic Functions, p. 57) states that we may put $$g_2 = \frac{1 - \lambda + \lambda^2}{12}, ...
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90 views

The periods of the Weierstrass function $\wp(z)$

Is it true that the periods $\omega_1$, $\omega_2$ of $\wp(z)$ are $\omega_1 = 4K$ and $\omega_2 = 4iK'$, respectively? Here, $K = K(k)$ is the complete elliptic integral of the first kind, and $K' = ...
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Zeros of Weierstrass p function

I would like to know where the zeros of the $\wp$ function lie in terms of its periods. I know that we can locate the zeros of its derivative, $\wp'$, but I can't figure how to locate the roots of the ...
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77 views

Changes of variables to get an Elliptic Integral of the First Kind

I'm working with a non-linear second order ODE which has an analytical solution in terms of the Jacobi elliptical function $sn(u|k^2)$. The equation is $y''=y(\gamma - \frac{y^2}{2})$ where $\gamma$ ...
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56 views

Mean value formula

Let $u(x)$ be an entire positive solution of the equation $$\Delta u - u = 0 ~~~on ~ R^{n} ~~n>1$$ (a) Can you find a mean value formula? (b) Let $\mu$ be a positive Radon Measure on the unit ...
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1answer
55 views

Elliptic functions surjective

Is it true that every nonconstant elliptic function $f:\mathbf{C}/\Lambda\rightarrow\mathbf{P}^1$ is surjective? (I take elliptic functions to be defined on the torus $\mathbf{C}/\Lambda$) For how is ...
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1answer
52 views

Find coordinate $y$ of an elliptic curve point

If I have an elliptic curve over a finite filed $F_p$ ($p$ is prime) defined as $$ y^2 \equiv x^3 + ax + b\pmod p,$$ such that $4a^2 + 27b^2 \neq 0$ and suppose I have only given the coordinate $x$, ...
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An algebraic relation between any two elliptic functions with the same periods

I'm sorry if my question is trivial. In his book "Elements of the Theory of Elliptic Functions", page 44, Akhiezer proves, that any two elliptic functions with the same periods are connected by an ...
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Jacobian elliptic functions regarded as functions of the modulus.

Could anyone provide any references where the properties of Jacobian elliptic functions (and their derivatives) regarded as functions of the modulus are discussed? Especially it would be interesting ...
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46 views

Derivative of Integral of (g) with g in the limit

I would like to evaluate the following: $$\frac{\partial }{\partial \beta }\int _0^{\cos ^{-1}(\beta )}\text{dx} \sqrt{\beta +\cos (x)}$$ given that $0\leq\beta\leq1$ basically I'd like to find ...
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If a function's zeroes are doubly periodic, must that function be elliptic?

Proposition: Let a meromorphic function $f(z)$ be such that $f(z)=0\Leftrightarrow z \in\Lambda=\{n\omega_1+m \omega_2, (n,m)\in \mathbb{Z}^2\}$. Except in the following exception, must $f(z)$ be ...
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1answer
58 views

Is this function familiar to anyone?

Consider $$f(z)=\sum_{w\in C}\frac{1}{z-w}$$ Where $C$ is the set of complex integers. What I would like to know is where can I find any information about this function (name perhaps). For instance, ...
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268 views

Solving Differential Equations theoretically and using matlab

i am trying to solve the initial value and elliptic boundary value problems below. but now i need some help solving them using matlab. for the elliptic problem, any method is ok, but for the initial ...
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370 views

Questions aobut Weierstrass's elliptic functions

from the wikipedia: == In terms of the two periods, Weierstrass's elliptic function is an elliptic function with periods ω1 and ω2 defined as $$\wp(z;\omega_1,\omega_2)=\frac{1}{z^2}+ \sum_{n^2+m^2 ...
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77 views

Public Key Scheme decryption. [closed]

You have been sent a message based on the following Public Key Scheme. 1) Bob chooses two large primes $\ p,q $ with $ p \equiv q \equiv 2 \pmod 3$ and computes $ n=pq. $ 2) Bob chooses integers $ e,d ...
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Expression of $2\int_{0}^{\frac{1}{r_{0}}}\frac{du}{\sqrt{\frac{r_{0}-r_{s}}{r_{0}^{3}}-u^{2}\left(1-u r_{s}\right)}}$ in terms of elliptic integrals

In Gravitation by Misner et al. 1973 the authors state that (calculus related to the Schwarzschild metric page 678) : ...
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84 views

Fundamental period of the WeierstrassP elliptic function?

Consider the WeierstrassP elliptic function $\wp(z, g_2, g_3)$ with the invariants $g_2\in\mathbb{R}$ and $g_3\in\mathbb{R}$: $$\wp'(z)^2 = 4\wp(z)^3 - g_2 \wp(z) - g_3$$ According to Wikipedia when ...
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136 views

From inverse Weierstrass function to Jacobi elliptic/inverse elliptic functions?

As a conclusion to a previous question on integrals, I get an answer in terms of inverse Weierstrass elliptic function : $$ f\left(x\right)=\wp^{-1}\left( \beta + \frac{9\beta^2-1}{3(x-\beta)} \right) ...