Questions tagged [elliptic-functions]
Questions on doubly periodic functions on the complex plane such as Jacobi and Weierstrass elliptic functions.
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On Ramanujan's fastest series.
Context
With some effort we can show that Ramanujan's fastest series implies:
\begin{align}
\frac{8E(k_{58})K(k_{58})}{\pi^2}-\frac{aK^2(k_{58})}{\pi^2}=\frac{\sqrt{58}}{29\pi},\tag{1}
\end{align}
...
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For which of the Weierstrass elliptic function periods do this equation of the modular discriminant and the Dedekind eta function apply?
It is often claimed that the following equation holds for the modular discriminant $\Delta=g_2^3-27g_3^2$ of the Weierstrass elliptic funtion and the Dedekind eta ($\eta$) function for period ratio $\...
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1
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The holomorphic differential of an elliptic curve as a Riemann surface
I am reading a Teichmueller theory book and trying to understand elliptic curves as examples of Riemann surfaces.
Consider the elliptic curve
\[X = \{[z : w : y] \in \mathbb C \mathrm P^2 \mid w^2y = ...
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1
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Close form expression for an integral with z derivative of jacobi theta function
I have an expression of the form
$$
\tag{1}
A(\chi) = \int_{0}^\infty\sum_{i=0}^\infty (-1)^{i+1}\frac{(2i+1)\chi}{\sqrt{t}}\exp\left(\frac{-(2i+1)^2\chi^2}{t}\right)\mathrm{d}t.
$$
If I am not ...
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Proof of ellipticity of lemniscate functions from integral definition
The lemniscate functions $\text{sl}$ and $\text{cl}$ are the solutions to the differential equation $$ (y')^2+y^4=1$$ with $\ y(0)=0, \ y'(0)=1$ $\ $ or $\ $ $y(0)=1, \ y'(0)=0.$
Using the integral ...
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Syntax of Jacobi elliptic functions
I'm trying to understand this paper on blackholes
https://articles.adsabs.harvard.edu/pdf/1979A%26A....75..228L
the following is in the paper:
picture of equations
However, I do not understand the ...
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Relationship Weierstrass elliptic function, Jacobi elliptic functions
On Wikipedia, one finds a relation between the Weierstrass elliptic function and Jacobi's elliptic functions as
$$
\wp(z; g_2; g_3) = e_3 + \frac{e_1 - e_2}{sn^2 w}
$$
where $e_1$, $e_2$, $e_3$ are ...
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Does the theory of theta functions ever go beyond just random caulculations and ugly formulas?
I recently took a class in elliptic functions and found the theory of elliptic functions very clean. You get very powerful results and the proofs are usually short and intuitive. Then we learned about ...
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inverse problem about scalar multiplication on koblitz curves (or more exactly the secp256k1)
My problem is given $Q=nP$ to find point $P$ given 257 bits long integer $n$ and point $Q$.
It’s something possible on other curves but Koblitz curves have extra characteristic and can’t be converted ...
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Proving the fundamental period of lemniscate sin and cos function is $\{(1+i)\varpi,(1-i)\varpi\}$
(I couldn't find many sources of the derivation on the internet, so I might as well show most of my work here)
I started by using the Argument sum formulas (see this post), which require specific ...
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1
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Pole expansion and Fourier series of lemniscate sine function
Given that
$$\frac{\varpi}{\text{sl}(\varpi z)}=\sum_{n,k\in\mathbb{Z}}\frac{(-1)^{n+k}}{z+n+ik}$$
It can be deduced that, for $-1<\text{Im}(z)<1$:
$$\frac{1}{\text{sl}(\varpi z)}=\frac{\pi}{\...
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Proving the Argument sum formula for lemniscate sine and cosine
Note that I take the Definition of sl (Lemniscate sine) and cl (Lemniscate cosine)
as the inverse of $\int_0^z \frac{1}{\sqrt{1-t^4}} dt$ and $\int_z^1 \frac{1}{\sqrt{1-t^4}} dt$.
Following from this ...
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Proving a connection noticed by Gauss between lemniscatic functions and spherical geometry.
In p.415 of volume 3 of Gauss's werke one can find the following remark of Gauss:
[Later note]:
I. $$\alpha+\delta+\gamma = \pi [=\varpi]$$
set $$\mathbb{sinlemn}(\alpha) = \mathbb{tang} (a), \...
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Reference request: Elliptic Integrals and Elliptic Functions
Elliptic integrals, elliptic functions, and elliptic curves are well-studied objects in mathematics. Unfortunately, for various reasons, they are usually not covered in undergraduate courses. I feel ...
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Real elliptic curve embedded into complex torus
Main goal: Visualize the "real slice" of elliptic curve from the complex torus perspective
Basic setup: for $\tau \in \mathbb H$ (the upper half plane $\{z\in \mathbb C: \Im(z)>0\}$), one ...
2
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0
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Elliptic functions by Eisenstein-Kronecker
In $\textit{Elliptic Fuctions according to Eisenstein and Kronecker}$, chap VIII, section 13 by A.Weil there is the following problem
For any integer $k \geq 0$ and $z, w \in \mathbb{C}$, the function ...
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Second order differential equation with function of Jacobi elliptic function as a solution
Many years ago, I remember reading in a book how to transform the following ODE
$$
f'' = af^3 -bf + c/f^3
$$
into a form that lead to a solution in terms of Jacobi elliptic functions. If I am not ...
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Integrals of (Jacobi) elliptic functions
Jacobi elliptic functions create a new trigonometry that we don't teach in high schools.
Taking the derivative of composite functions of these functions with elementary functions is easy thanks to ...
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Second order homogeneous ODE with Jacobi elliptic coefficient
I am stuck looking for a solution for the 2nd order ODE
$$ x''(t) + \omega (t) x(t) =0$$
with $ \omega(t)=-1+3(1-e) \, nd \left( \sqrt{\frac{1+e}{2}} t, \ \frac{2e}{1+e} \right)^2$, where $nd$ is the ...
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Is $\int \operatorname{sn}^2u\,\mathrm du$ really irreducible?
Let $\operatorname{sn}$, $\operatorname{cn}$, $\operatorname{dn}$ be Jacbian elliptic functions (https://dlmf.nist.gov/22). According to Greenhill,
The integrals
$$\operatorname{sn}^2u,\operatorname{...
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Representation of Weierstraß $\wp$ function for $\Lambda=\Bbb Z+\Bbb Z i$ as series over trigonometric function
The Weierstraß $\wp$ function for a lattice $\Lambda\subset\Bbb C$ can be defined by the sum
$$
\wp(z) = \frac1{z^2} ~+\!\! \sum_{\lambda\in\Lambda\setminus\{0\}}
\left(\frac1{(z-\lambda)^2}-\frac1{\...
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1
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Values for which $\frac{K(k')}{K(k)}=a+i$ where $a$ is an algebraic number.
Context
Being:
$$K(k)=\int_{0}^{\pi/2}\frac{dx}{\sqrt{1-k^2\sin^2{x}}},\tag{1}$$
the complete elliptic integral of the first kind, and
$$K(k')=\int_{0}^{\pi/2}\frac{dx}{\sqrt{1-k'^2\sin^2{x}}},\tag{2}$...
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1
answer
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Proof that $\zeta (z)=\frac{\sigma '(z)}{\sigma (z)}$
The Weierstrass $\sigma$ function of a lattice $\Omega$ is defined by
$$\sigma (z)=z\prod_{\omega\in \Omega\setminus\{0\}}\left(1-\frac{z}{\omega}\right)\exp\left(\frac{z}{\omega}+\frac{1}{2}\frac{z^2}...
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2
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Proving $\sum_{n=-\infty}^\infty n^2e^{-\pi n^2}=\frac{\Gamma (1/4)}{4\sqrt{2}\pi^{7/4}}$
I conjecture that
$$\sum_{n=-\infty}^\infty n^2e^{-\pi n^2}=\frac{\Gamma (1/4)}{4\sqrt{2}\pi^{7/4}}$$
because the left-hand side and right-hand side agree to at least $50$ decimal places. Is the ...
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Weierstrass sigma function identity - Silverman AEC 6.3
In Silverman's AEC Chapter VI we define the Weierstrass $\wp$ and $\sigma$ functions, particularly for $\Lambda = \mathbb{Z} + \tau \mathbb{Z}\subset \mathbb{C}$ a lattice, $$\wp(z) = \wp(z,\Lambda) =...
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What is the range of the Weierstrass elliptic function?
There is the following function:
$$
\wp(z,\Lambda):=\frac{1}{z^2} + \sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac 1 {(z-\lambda)^2} - \frac 1 {\lambda^2}\right)
$$
What values can it take? For me ...
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Minimal polynomial of $\operatorname{cd}\frac{4K}{n}$
Define
$$K=\int_0^1 \frac{dx}{\sqrt{1-x^4}}$$
and the Jacobi elliptic function $\operatorname{cd}$ with modulus $i$ by
$$\int_{\operatorname{cd}z}^1\dfrac{dx}{\sqrt{1-x^4}}=z$$
on $z\in [0,2K]$ and by
...
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2
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Given algebraic $a$, find the closed form of $\int_0^a \dfrac{dx}{\sqrt{1-x^4}}$
Let
$$A=\int_0^1 \dfrac{dx}{\sqrt{1-x^4}}.$$
Given an algebraic number $0\le a\le 1$, can we determine if there exists a rational number $b$ such that
$$\int_0^a \dfrac{dx}{\sqrt{1-x^4}}=Ab?$$
If so, ...
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Mapping the upper half-plane onto rhombus
Book's question: Map the upper half-plane $\Im z>0$ onto a rhombus in the $w$-plane with angle $\alpha\pi$ at the vertex $A=0$ and side $d$. The correspondence of the points is given by $A=0\to z=0$...
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Connection between Lame equation with Weierstrass and elliptic sine
Lame function is the solution of the following equation,
$$\frac{d^2w}{dz^2}+\left(A+B\wp(z)\right)w=0,$$
where $A$ and $B$ are constants and $\wp(z)$ is the Weierstrass elliptic function. Wiki says ...
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Prove that the Weiertrass function is biperiodic [duplicate]
I am reading the chapter on elliptic functions in complex analysis by ahlfors, who used the following formula in his argument that the Weiertrass function is biperiodic
We let
$$\wp=\frac{1}{z^2}+\...
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Transcendence of periods of the Weierstrass elliptic function
In the following, $\wp$ denotes the Weierstrass elliptic function (https://en.wikipedia.org/wiki/Weierstrass_elliptic_function), $g_2$ and $g_3$ are invariants and $\omega_1$ and $\omega_2$ are ...
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Inversion of the imcomplete elliptic integral of 2nd kind: $E(z|m)=x\;\Longleftrightarrow\; z=\text{bm}(x|m)$
I would like an opinion on this issue: similarly to the fact that
$$F(z|m)=x\Longleftrightarrow z=\text{am}(x|m)$$
I also tried to reverse the function $E(z|m)$ and got the following series (I ...
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Found the parameter of the elliptic integral of the first kind
Suppose I have:
$$\frac{p}{[K(p)-K(\frac{-\pi}{2},p)]^2} = x$$
$K(p)$ being the complete elliptic function of the first kind and $K(\theta,p)$ the incomplete elliptic function of the first kind.
How ...
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Elliptic function of the third kind $\Pi(n|m)$ for $n>1$
Can someone help me with the following problem?
I have the complete elliptic integral of the 3° kind defined as follow:
$$\Pi(n|m):=\int_0^{\frac{\pi}{2}}\frac{\mathrm{d}t}{(1-n\sin(t)^2)\sqrt{1-m\sin(...
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Trasformation of elliptic integral $\Pi(n;x|m)$ in function of $F(x|m)$ and $E(x|m)$
I everyone,
I calculated these two formulas for $m<1$ and $z\in\mathbb{C}$:
$$K(m)=F\left(\left.\frac{\pi}{2}-z\right|m\right)+\frac{1}{\sqrt{1-m}}\cdot F\left(z\left|\frac{m}{m-1}\right.\right)$$
$...
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More on Eisenstein-like series.
Context:
This post is related to Conjectured closed forms for Eisenstein-like series . This is an extension of it.
We have also:
\begin{align*}
S(x):=\sum_{n=1}^{\infty}\frac{(2n-1)\left(e^{\frac{\pi(...
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Rutherford constant and integrals involving elliptic functions: $\displaystyle\int_{0}^{1}\frac{E(x)^2}{K(x)}\mathrm{d}x$
Introduction
I found this definition of the Rutherford constant on Wolfram's site:
$$\mathcal{K}_R=\sqrt{2}\int_{-1}^{\infty}\frac{R(x)^2}{S(x)}\mathrm{d}x$$
Where
$$R(x)=\frac{1}{2\pi}\int_{\overline{...
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Higher-order 'clover functions' are not elliptic--how to see this?
Define the '$n$-clover function' $ \phi _n $ as the inverse of the function
$$ f_n(r) := \int_0^r \frac{dx}{\sqrt{1-x^n}} $$
This is related to studying the arc length of clover curves (among other ...
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Conjectured closed forms for Eisenstein-like series
This question is related to:
Eisenstein sum.
Being $q=e^{\pi}$, we have also:
\begin{align*}
\sum_{n=1}^{\infty}\frac{n(q^{n}(-1)^{n}+1)}{q^{2n}+2(-1)^{n}q^{n}+1}=-\frac{1}{24}\tag{1},
\end{align*}
\...
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0
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Conformal mapping from triangle to upper half-plane
I try to understand the answer to the following question as I want to deepen my knowledge about conformal mapping: Conformal mapping from triangle to upper half plane.
I do not have enough knowledge ...
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0
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Solving a system of integral equations of the form $\int_0^r K(r,s)f(s)\mathrm{d}s + \int_r^1 Q(r,s)g(s) \mathrm{d}s = 1$ with $s,r\in[0,1]$
Consider the following system of integral equations for the unknown functions $A(s)$ and $B(s)$,
\begin{align}
\int_0^r \left( \phi_1 \left(\tfrac{s}{r} \right) A(s) + \phi_2\left(\tfrac{s}{r} \right)...
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Exercise 12 in Ch. 1 of Apostol's Modular Functions and Dirichlet Series - How to See The Vanishing of Eisenstein Series
The mentioned problem has you prove $$G_{2k}(\frac{-1}{\tau}) = \tau^{2k}G_{2k}(\tau).$$ From there, it asks that you deduce the following,
$$G_{2k}(e^{2 \pi {i} / 3})=0$$
if $k \neq 0 \mod 3$. I see ...
26
votes
2
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A Tough Series: $\sum_{n=0}^\infty2^na_{n+1}\sqrt{a_n}=\frac{\Gamma^2(\frac14)-4\cdot\Gamma^2(\frac34)}{8\sqrt{2\pi}}$
If $\displaystyle a_0=\frac12$ and $\displaystyle a_{n+1}=\frac{1-\sqrt{1-a_n}}{1+\sqrt{1-2a_n}}$, show that
$$\sum_{n=0}^\infty2^na_{n+1}\sqrt{a_n}=\frac{\Gamma^2(\frac14)-4\cdot\Gamma^2(\frac34)}{8\...
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0
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Confusion of Generator point in it's Montgomery Form and Weierstrass Form for secp256k1
I am using GEC Module (https://github.com/HareInWeed/gec) to perform point operations on secp256k1. Here, the generator point is defined as below
...
5
votes
2
answers
353
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Sum involving the Euler's Pentagonal Number function derivative.
Being:
$$\sum_{n=-\infty}^{\infty}(-1)^nq^{n(3n-1)/2}(3n^2-n) \tag{1},$$ it can be shown:
$$\sum_{n=-\infty}^{\infty}(-1)^nq^{n(3n-1)/2}(3n^2-n)=-\frac{1}{12}(1-P(q))\prod_{n=1}^{\infty}(1-q^n)\tag{2},...
4
votes
1
answer
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Finding a concise relation between $\operatorname{ns}\left(\frac{K(k)}{3},k\right)$ and $\operatorname{ns}\left(\frac{2K(k)}{3},k\right)$
Let $\operatorname{ns}\left(z,k\right)$ be one of the Jacobi's elliptic functions, and $K(k)$ the complete elliptic integral of the first kind. It's well-known that $\operatorname{ns}(mK(k),k)$ where $...
4
votes
3
answers
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How to prove the infinite product expression of this Jacobian elliptic $\operatorname{sn}$ function?
How to prove the following infinite product expression?
\begin{align*}
\operatorname{sn}(x,k)=\tanh \left(\frac{\pi\,\!x}{2 K(k')}\right)\prod\limits_{n=1}^{+\infty\,}\frac{ \tanh \left(\frac{\pi\left(...
2
votes
1
answer
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Eisenstein sum.
I have a proof of:
$$S=\sum_{n=1}^{\infty}\frac{n(-1)^{n+1}}{(-1)^n+e^{\pi n}}=\frac{1}{24}.$$
That is related to Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}...
2
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0
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Elliptic-Function Radical $R = x\sqrt{x\sqrt{\sqrt{x\sqrt{\sqrt{\sqrt{x...}}}}}}$
This post concerns a radical I included in an earlier post that didn't receive much attention, and I hypothesize that this radical has quite an interesting closed form, therefore I'm posting this.
(...