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

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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
36 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
73 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
255 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
75 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
21 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
22 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
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2answers
99 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
120 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
119 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
57 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 ...
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3answers
227 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 ...
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112 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 ...
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1answer
57 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 ...
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1answer
38 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 } ...
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1answer
59 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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3answers
151 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 ...
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1answer
106 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, ...
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2answers
69 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 ...
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1answer
98 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 $$J(\tau)=\frac{4}{27} \frac{(1-\lambda+\lambda^2)^3}{\lambda^2(1-\lambda)^2} $$ ...
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1answer
59 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 $\text{Im} \tau > 0$. ...
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1answer
42 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 ...
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1answer
49 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 ...
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1answer
159 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 ...
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1answer
72 views

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, ...
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2answers
442 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 ...
3
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2answers
182 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 ...
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2answers
469 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?
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1answer
93 views

Definition Weierstrass $\zeta$-function unclear

The Weierstrass $\zeta$-function is defined as follows for a lattice $\Lambda$, where a lattice is a discrete subgroup of $\mathbb{C}$ containing an $\mathbb{R}$-basis for $\mathbb{C}$. $$\zeta(z) = ...
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1answer
155 views

Can we express all doubly periodic functions as one of doubly periodic function?

Singly Periodic Functions $e^{x},\cos(x),\sin(x),\tan(x), .. etc.$ Euler's identity is $$e^{i\alpha}=\cos(\alpha)+i\sin(\alpha)$$ $$e^{-i\alpha}=\cos(\alpha)-i\sin(\alpha)$$ Thus, we can express ...
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2answers
167 views

Subtraction two points on elliptic curve.

Suppose Q, T and S are three points on an elliptic curve, such that Q+T = S. With knowing Q and S, can we compute T? In other word whether exists subtraction operation on elliptic curve, or not?
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1answer
147 views

How to prove that $\frac{\eta^{14}(q^4)}{\eta^{4}(q^8)}=4\eta^4(q^2)\eta^2(q^4)\eta^4(q^8)+\eta^4(q)\eta^2(q^2)\eta^4(q^4)$?

How can we prove that $$\frac{\eta^{14}(q^4)}{\eta^{4}(q^8)}=4\eta^4(q^2)\eta^2(q^4)\eta^4(q^8)+\eta^4(q)\eta^2(q^2)\eta^4(q^4) \ ?$$ Here, $\eta(q)$ is the Dedekind Eta Function, which is defined by ...
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1answer
102 views

Addition formulas for Jacobi amplitude function

Are there any known summation formulas for the Jacobi amplitude function? I need a formula like $\mathrm{am}(t+x)=\mathrm{am}(t) + f(x)$. I have plotted some graphs and it seems that $f(x)$ is ...
0
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1answer
44 views

Direct construction of an arbitrary elliptic function of order $2$ with pole set contained in its lattice.

So I was asked to prove that every Elliptic function of order $2$ whose pole set is contained in the lattice $\Lambda$ is of the form $a+b\wp$, where $\wp$ is the Weierstrass-p function: ...
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3answers
191 views

Translation of Weber's Lehrbuch der Algebra vol 1, 2, 3

I have been trying to study elliptic functions and theta function for quite some time and have already got the hang of the classical theory (Jacobi/Ramanujan) based on real analysis, and now would ...
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3answers
252 views

Can you recommend some books on elliptic function?

I plan to study elliptic function. Can you recommend some books? What is the relationship between elliptic function and elliptic curve´╝čMany thanks in advance!
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4answers
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Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$

I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$. I found a solution by myself 10 hours after I posted it, here it is: ...
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2answers
78 views

Proof of $\left| \operatorname{cn}\left( x(1+ i) \mid m \right)\right|=1$ for $m=\frac{1}{2}$ and $x \in \mathbb{R}$

Let $\operatorname{cn}(u\mid m)$ be Gudermann's notation for the Jacobi elliptic function $\operatorname{cn}$. It is well known to be doubly periodic. For $0<m<1$ the two periods, $4 K(m)$ and ...
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1answer
223 views

Numerical values of the Jacobi elliptic function sn: Wolfram Alpha vs. Maple vs. C++?

I have a problem to check the validity of an algorithm I've implemented in C++ to compute the Jacobi elliptic function $\mathrm{sn}(u, k)$ (inspired and improved from Numerical Recipes 3rd edition). I ...
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133 views

General solution of a differential equation $x''+{a^2}x+b^2x^2=0$

How do you derive the general solution of this equation:$$x''+{a^2}x+b^2x^2=0$$where a and b are constants. Please help me to derive solution thanks a lot.
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Semiperiod of $\wp$

If we let $u$ and $v$ span a lattice of $\mathbb{C}$, then for the associated $\wp$-function: $\wp(\frac{u}{2})$ is algebraic over $\mathbb{Q}(G_k(u,v),k \ge 4)$ because it is a zero of $$4X^3 - ...
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2answers
158 views

How do you solve an equation involving the Weierstrass p-function?

What $z\in \mathbb{C}$ satisfy \begin{eqnarray*} \wp(z) = \wp(nz) \end{eqnarray*} if $n\in \mathbb{N}$? The Weierstrass p-function is defined as \begin{eqnarray*} \wp(z)=\frac{1}{z^2}+\sum_{\omega\in ...
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51 views

Even elliptic function

If I have an elliptic function $f$ with respect to some lattice $L$, and $f$ has poles in the points of $L$ and nowhere else, does it follow that $f$ is even? I think so, and my idea was to write $f ...
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142 views

Analytic functions can't have more than two periods

Let $f(z)$ be a non-constant analytic function. Show that $f(z)$ can't have more than two periods.
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References for the conformal equivalence of the space of complex 1-tori and C?

What are some good references with proofs of the conformal equivalence of the space of complex tori and $\mathbb{C}$? So far I only have the book by Jones and Singerman.
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294 views

Turning an elliptic curve over C into a complex torus

I have been reading a lot about the Weierstrass $\wp$ function and I understand the parameterization of an elliptic curve with the elliptic function( i.e. $x=\wp(z)$ and $y=\wp^\prime(z)$). I would ...
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1answer
109 views

Direct proof of the non-zeroness of an Eisenstein series

Question: Can you show directly from its formula that $G_4(i)\neq0$? Recall that the holomorphic Eisenstein series of weight $2k$ is defined by: $$G_{2k}(\tau)= \sum_{(m,n)\in\mathbb{Z}^2\setminus ...
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Question about constructing the Weierstrass $\wp$ function [duplicate]

Possible Duplicate: convergence of a particular series I am reading about the construction of the Weierstrass $\wp$-function for an arbitrary lattice in $\mathbb{C}$. If $a, b$ are complex ...
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0answers
90 views

Elliptical Integrals and graphing plot

I'm trying to computer plot the graphs of sn(u), cn(u) and dn(u) for k = 1/4, 1/2, 3/4, 0.9 and 0.99 And I am trying to plot 3D graphs of sn, cn and dn as functions of u and k. Here's what I have: ...
7
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1answer
156 views

Finding x in $\frac{\,_2F_1(\frac{1}{5},\frac{4}{5},\,1,\,1-x)}{\,_2F_1(\frac{1}{5},\frac{4}{5},\,1,\,x)} = \sqrt{n}$

I was trying to find a closed-form for $0<x<1$ in, $$\frac{\,_2F_1(\frac{1}{m},\,1-\frac{1}{m},\,1,\,1-x)}{\,_2F_1(\frac{1}{m},\,1-\frac{1}{m},\,1,\,x)} = \sqrt{n}$$ where $\,_2F_1(a,b,c,z)$ ...