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

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32 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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9 views

$\exists C \in \mathbb{C}^{\times}$, $n\in\mathbb{N}$: $f = C\Delta^n$

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
23 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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19 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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39 views

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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0answers
43 views

When does $\wp$ take real values?

Whittaker and Watson mention that when the invariants of Weierstrass $\wp$ function are such that $g_2^2 - 27g_3^2 > 0$, and if $2\omega_1$ and $2\omega_2$ are its periods then the function takes ...
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1answer
26 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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2answers
47 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
13 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
25 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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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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28 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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25 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 ...
3
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1answer
60 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 ...
5
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1answer
48 views

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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33 views

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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1answer
88 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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1answer
78 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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36 views

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 ...
3
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1answer
65 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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51 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
38 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
44 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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55 views

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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40 views

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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2answers
43 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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34 views

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
55 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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1answer
227 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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2answers
200 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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1answer
73 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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29 views

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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1answer
73 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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107 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) ...
2
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1answer
109 views

How were the solutions to these differential equations found?

These two very strange differential equations came up yesterday while I was doing a physics problem that I made up: EQ 1) $y'^{2} = k \sin(y)$ EQ 2) $y'' = k\cos(y)$ where $y'$ means ...
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1answer
78 views

Question about divisors

Let $\Lambda=<2\omega_1,2\omega_2>$ be a lattice in $\mathbb{C}$, $C=\mathbb{C}/\Lambda$ an elliptic curve. Let $O(0,0)$ (neutral element of the lattice), and $A \in \mathbb{C}/\Lambda$ point of ...
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2answers
95 views

Constructing a meromorphic function

I need help with the following problem. "Let $C : y^2 = x^3 − 5x^2 + 6x$ be a cubic curve with the standard group law. Find a meromorphic function on $C$ having the pole of order two at ...
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1answer
56 views

How to obtaining the lattice corresponding to an elliptic curve

Let $C$ be a complex elliptic curve given by the quation $y^2=4x^3-g_2 x -g_3$. How do I find the lattice $\Lambda$ such that $C \cong \mathbb{C}/\Lambda$? I need the lattice (and corresponding ...
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0answers
117 views

Coefficients of the Weierstrass $\wp$'s Laurent expansion

I am trying to study the Weierstrass $\wp$ function using a combination of texts from Alfors; Cartan; Freitag and Busam; and Siegel. But I am having some trouble because I would like to try to avoid ...
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1answer
63 views

The Jacobi nome $q$

Does anyone know why $q = e^{-\pi K'/K} = e^{\pi i \tau}$ is called the nome? Is there a historical reason? Does the word nome mean something in Latin or German?
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1answer
362 views

Finding points on an elliptic curve

I have an elliptic curve $$x^3+17x+5 \mod 59$$ $P = (4,14)$ is given and I need to find point $8P$. to calculate $8P$, I first calculated $2P$ by using the equation sigma = 3x^2+a/2y = ...
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1answer
485 views

Summation over Weierstrass $\wp$ functions

I've been trying to prove the following closed expression for a summation over Weierstrass $\wp$-functions: \begin{equation} \sum_{k=1}^{N-1} \wp_N(k) = ...
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1answer
119 views

The transformations on the nome and Landen's transformation

Could someone please explain how to transform the nome $q = e^{-\pi K'/K}$ from $q^2$ to $q$ and then to $-q$? In other words, how does changing $q^2$ to $q$ and then $q$ to $-q$ affect $k$ and $K$. ...
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2answers
75 views

A general solution to the sine-gordon equation (with only time dependence)

I have a non-linear equation similar to the sine-gordon equation, but it is ordinary: $\frac{d^2x}{dt^2} - g \sin(x) = 0$ where $g$ is some positive constant. I'm looking for a general solution of ...
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0answers
43 views

Inverse of an arbitrary elliptic integral

Given an arbitrary elliptic integral $I\left( x \right) = \int_{c}^{x} R \left(t, \sqrt{P(t)} \right) \, dt,$ where $R$ is a rational function of its arguments and $P\left( t \right)$ a polynomial ...
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2answers
164 views

Hypergeometric formulas for the j-function

Given the j-function $j(\tau)$ and elliptic lambda function $\lambda(\tau)$. Define, $$g = -1+2\frac{\,_2F_1\big(\tfrac{1}{2},\tfrac{1}{2},1,\,\lambda(2\tau)\big) ...
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1answer
362 views

Heuman Lambda Function in MATLAB

I'm trying to implement the Heuman Lambda function in MATLAB, but i'm having problems getting a correct answer. The Heuman Lambda Function is: $$ \Lambda_0(\beta,k) = \frac{2}{pi}[E(k)F(\beta,k') + ...
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1answer
155 views

Weierstrass $\wp$-function: $(\partial_z \wp(z,\omega))^2$

Let $\vartheta(z,\omega)$ be the Riemann theta function. For $j \in \mathbb{Z}$ let $c_j$ be the coefficient of $z^{j}$ in the Laurent expansion of $\partial_z \log \vartheta \left(z + \frac{1 + ...
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1answer
73 views

Embedding of elliptic curves into $\mathbb{P}^2$ by arbitrary line bundle of degree $3$

Let $E$ be a complex elliptic curve, with distinguished point $x_0 \in E$. Any divisor of degree three is equivalent to the divisor $D=x+2x_0$. If $x=x_0$, it is well known that $L(D)$ has an explicit ...