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

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4
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1answer
80 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}, ...
3
votes
1answer
65 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' = ...
3
votes
0answers
24 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
votes
1answer
58 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 $\gamma$ ...
0
votes
0answers
46 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 ...
1
vote
1answer
26 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 ...
1
vote
1answer
36 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 $x$, ...
0
votes
0answers
44 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 ...
1
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0answers
34 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 ...
0
votes
2answers
41 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 ...
1
vote
0answers
26 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 ...
1
vote
1answer
52 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, ...
1
vote
1answer
167 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 ...
3
votes
2answers
151 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 ...
0
votes
1answer
68 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 ...
2
votes
0answers
27 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) : ...
4
votes
1answer
65 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 ...
2
votes
0answers
70 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
votes
1answer
106 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 ...
2
votes
1answer
74 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 ...
1
vote
2answers
77 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 ...
3
votes
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 ...
2
votes
0answers
87 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 ...
1
vote
1answer
58 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?
1
vote
1answer
188 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 = ...
9
votes
1answer
450 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) = ...
1
vote
1answer
111 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 $K$. ...
0
votes
2answers
62 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 ...
1
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0answers
35 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 ...
2
votes
2answers
154 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) ...
1
vote
1answer
297 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') + ...
2
votes
1answer
143 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 + ...
3
votes
1answer
66 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 ...
6
votes
3answers
506 views

Solving 5th degree or higher equations

According to this, there is a way to solve fifth degree equations by elliptic functions. Some related questions that came to mind: Besides use of elliptic functions, what other (known) methods are ...
8
votes
0answers
154 views

Global sections of vector bundles on a complex elliptic curve and analytic functions

Let me fix an elliptic curve $E$ over complex numbers with distinguished point $x \in E$. Thanks to Atiyah we know everything about discreet parameters of vector bundles and its moduli spaces. But I ...
1
vote
1answer
93 views

The natural boundary of the modular $\lambda$ function is the real axis

I want to solve the following problem from Ahlfors' text: Show that the function $\lambda(\tau)$ introduced in Chap. 7, Sec. 3.4, has the real axis as a natural boundary. Here $\lambda(\tau)$ is ...
1
vote
1answer
54 views

Expansion of Jacobi's $\theta_3(0,q)$ in q=1

In trying to solve a certain limit, I wondered how Mathematica comes up with this weird expression for a series expansion of Jacobi's $\theta_3(0,q)$ in $q=1$ at the order 0: $\frac{i \sqrt{\pi } ...
0
votes
1answer
106 views

Generalized argument principle and elliptic functions

This is a homework question, so please only hints / suggestions if at all possible. The question asks If $f$ is an elliptic function with respect to some lattice $\Lambda$ and $z_1, \dotsc, z_k ...
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votes
3answers
170 views

Showing Weierstrass elliptic function has periods

Let $$\wp(z)=\frac{1}{z^2}+\sum_{w \in \Lambda^*} \left[\frac{1}{(z+w)^2}-\frac{1}{w^2}\right] $$ be the Weierstrass elliptic function with $\Lambda=\Bbb{Z}+\Bbb{Z}\tau$, $\Lambda^*=\Lambda-0$. I ...
0
votes
1answer
179 views

Orders of poles/zeros of an even elliptic function

I am reading a proof of the fact that every even elliptic function $f$ with periods $1$ and $\tau$ is a rational function of the Weierstrass $\wp$ function. The proof seems to use this fact often, ...
2
votes
2answers
102 views

Power Series of Elliptic Functions

I need some help for the following problem: Obtain the following power expansions for the elliptic functions: $$sn(u)=u-{(1+k^2) \over3!}u^3+{(1+14k^2+k^4) \over 5!}u^5-...$$ $$cn(u)=1-{1 ...
4
votes
1answer
136 views

Where does Klein's j-invariant take the values 0 and 1, and with what multiplicities?

I tried to solve the following problem from Ahlfors' text, please verify my solution: Where does the function $$J(\tau)=\frac{4}{27} \frac{(1-\lambda+\lambda^2)^3}{\lambda^2(1-\lambda)^2} $$ ...
3
votes
1answer
70 views

Why is the modular $\lambda$ function a quotient of two meromorphic functions in the U.H.P.?

In Ahlfors' complex analysis text, page 278 it says: It is quite clear from (9) that $\lambda(\tau)$ is the quotient of two analytic functions in the upper half plane $\text{Im} \tau > 0$. ...
0
votes
1answer
56 views

Finding maximum of a function with an ellipse constraint

I'm trying to find the maximum of a function $f = a^T\mu$ where a is a vector when we have a constraint of the form: $$g(\mu) = n\mu^T S^{-1}\mu - C = 0$$ where C is a fixed constant, $S^{-1}$ is a ...
2
votes
1answer
64 views

Does the Weierstrass $\wp$ function have any double values besides $\infty$?

Given nonzero complex constants $\omega_1,\omega_2$, with nonreal ratio, we define $$\wp(z;\omega_1,\omega_2)=\frac{1}{z^2}+ \sum_\omega \frac{1}{(z-\omega)^2}-\frac{1}{\omega^2} $$ where the sum is ...
2
votes
1answer
287 views

Any even elliptic function can be written in terms of the Weierstrass $\wp$ function

Given two nonzero complex numbers $\omega_1, \omega_2$, with nonreal ratio, we define the period module $$M= \omega_1 \mathbb Z+ \omega_2 \mathbb Z= \{n_1 \omega_1+ n_2 \omega_2:n_1,n_2 \in \mathbb Z ...
3
votes
1answer
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I can't understand why Ahlfors' statement is true (isolation of singular points)

In Ahlfors' complex analysis text, page 264, he writes: When $\Omega$ is the whole plane $F(\zeta)$ has isolated singularities at $\zeta = 0$ and $\zeta=\infty $. Reading the previous sections, ...
21
votes
2answers
480 views

Conjecture: $\int_0^1\frac{3x^3-2x}{(1+x)\sqrt{1-x}}K\big(\!\frac{2x}{1+x}\!\big)\,dx\stackrel ?=\frac\pi{5\sqrt2}$

$$\int_0^1\frac{3x^3-2x}{(1+x)\sqrt{1-x}}K\left(\frac{2x}{1+x}\right)\,dx\stackrel ?=\frac\pi{5\sqrt2}$$ The integral above comes from the evaluation of the integral ...
4
votes
2answers
285 views

How to transform a general higher degree five or higher equation to normal form?

This is a continuation of How to solve fifth-degree equations by elliptic functions? How to transform a general higher degree five or higher equation to normal form? For xeample, a quintic equation ...
17
votes
2answers
889 views

How to solve fifth-degree equations by elliptic functions?

From F. Klein's books, It seems that one can find the roots of a quintic equation $$z^5+az^4+bz^3+cz^2+dz+e=0$$ (where $a,b,c,d,e\in\Bbb C$) by elliptic functions. How to get that?