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Questions tagged [elliptic-functions]

Questions on doubly periodic functions on the complex plane such as Jacobi and Weierstrass elliptic functions.

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Integrate $\int f(x) sn(x,m) dx$ where $f(x)$ is a polynomial

To find the antiderivative of a product of a polynomial $f(x)$ and a sine function we can use this formula: $$\int f(x) sin(x) dx = F'(x) sin(x) - F(x) cos(x) + C $$ where $F(x) = f(x) - f''(x) + f^{(...
Loading - 146 Complete's user avatar
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Evaluate $\operatorname{nd}\left ( \frac{2K_6^\prime}{3},k_6^\prime \right ) \operatorname{sd} \left ( \frac{2K_6^\prime}{3} ,k_6^\prime \right )$

Could we get a simple expression for $$ \operatorname{nd}\left ( \frac{2K_6^\prime}{3},k_6^\prime \right ) \operatorname{sd} \left ( \frac{2K_6^\prime}{3} ,k_6^\prime \right ), $$ $$ k_6^\prime=\sqrt{...
Setness Ramesory's user avatar
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Excercise on j-invariant

I have this problem. Given $j:\cal{H}\rightarrow\mathbb{C}$ the j-invariant function defined on the upper-half complex plane as $j=1728\frac{g_2(\tau)^3}{g_2(\tau)^3-27g_3(\tau)^2}$, where $g_2,g_3$ ...
cespun's user avatar
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Exercise 2 in Chapter 1 of Apostol's "Modular Functions and Dirichlet series in Number Theory"

I am reading the book in the title and I don't know how to prove the following (that is Exercise 2 in Chapter 1): Suppose $f$ is an elliptic function (meaning that $f$ is meromorphic and there are two ...
Math101's user avatar
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5 votes
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205 views

How to prove the result of the following integral? [duplicate]

How to prove that $$ \int _0^{\infty }\frac{K\left(\frac{1}{2}-\frac{1}{2 \sqrt{x+1}}\right)}{\sqrt[4]{x+1}}e^{-x}{\rm d}x = \frac{1}{2} \sqrt{e \pi } K_{1/4}\left(\frac{1}{2}\right) $$ where $K(x)$ ...
Jie Zhu's user avatar
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How would you proceed with this integration question?

Q1: I am tasked to integrate the complete elliptic integral of the first kind $K(k)$ $$\int_{0}^{1}\frac{dk}{k'K(k)^2}$$ where k is the elliptic modulus. From this answer by Setness Ramesory you can ...
ProtoZone's user avatar
3 votes
3 answers
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Derive the use of the Jacobi Amplitude function with the nonlinear pendulum diffequation

I've been playing around with the pendulum differential equation ${\theta}''+\frac{g}{L}\sin{\theta}=0$, and have found many general solutions using the Jacobi Amplitude function $\text{am}(u,m)$ ...
Kareem Shamma's user avatar
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A Complex Analysis Problem About Weierstrass's Elliptic function [duplicate]

The question comes from A Course of Modern Analysis (E. T. Whittaker, G. N. Watson).Fifth Edition pages 476,which confused me for a long time.Only recently have I found time to rethink this question....
Ipigandothergod's user avatar
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A Complex Analysis Problem About Weierstrass's Elliptic function

The question comes from A Course of Modern Analysis (E. T. Whittaker, G. N. Watson).Fifth Edition pages 476,which confused me for a long time.Only recently have I found time to rethink this question....
Ipigandothergod's user avatar
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1 answer
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Derivatives of Weierstrass $\wp$ function .

I am working on a problem where I have to show that $\wp''$ is a quadratic in $\wp$. I know the the differential equation that $\wp$ satisfies. From there after taking derivatives of both of the sides ...
Sagnik Dutta's user avatar
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1 answer
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Proof that the Weierstrass $\wp$-function is of order $2$

Let $L$ be a lattice generated by $\omega_1, \omega_2$, then I should prove that the elliptic Weierstrass $\wp-$function $$\wp(z):=\frac{1}{z^2}+\sum_{l \in L \setminus \{ 0 \}}{ \left(\dfrac{1}{(z-l)^...
The Lion King's user avatar
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Why is the Jacobi amplitude real when $k>1$?

I understand the Jacobi amplitude is defined as the inverse function of the incomplete elliptic integral of the first kind $$ \mathrm{am}(u,k) = F^{-1}(u,k), $$ where $$ F(u,k) = \int_0^u \frac{d\phi}{...
Khalid Wenchao Yjibo's user avatar
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How to prove this peculiar relationship between minimal polynomials of Ramanujan class invariants?

The Ramanujan class invariants (a.k.a. "Ramanujan-Weber class invariants") are defined for $n>0$ by $$G_n=2^{-1/4}e^{\pi\sqrt{n}/24}\prod_{k=0}^\infty \left(1+e^{-(2k+1)\pi\sqrt{n}}\right)...
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Examples of non-elliptic Doubly Periodic Functions

I'm reading something on elliptic functions and I do not understand why in various textbooks the authors do not give non-trivial examples of doubly periodic and elliptic functions. Indeed, they only ...
TheWanderer's user avatar
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How do we prove that these series converge [closed]

I found some interesting series ,for some examples $$ \sum_{n=0}^{\infty}{\left( -1 \right) ^n\mathrm{arc}\tan \left( \tanh \left( \left( 2n+1 \right) \pi \right) \right)}=\frac{1}{2}\mathrm{arc}\tan \...
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Residue theorem and theta function identities

Let's use the classical definition $$ \vartheta _1\left( z,q \right) =-i\sum_{n\in \mathbb{Z}}{\left( -1 \right) ^nq^{\left( n+\frac{1}{2} \right) ^2}e^{i\left( 2n+1 \right) z}}\,\,\,\,\,\ q=e^{i\pi \...
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Growth rate of higher order derivatives of a variant of the modular lambda function on the imaginary upper half line

Let $f(z)=\lambda(-\frac{1}{1+z})=\frac{1}{1-\lambda(z)}$, where $\lambda(z)$ is the modular lambda function, $\lambda(z)=16q-128q^2+704q^3+\ldots$, where $q=e^{\pi i z}$ (https://en.wikipedia.org/...
user's user avatar
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What is the value of weierstrass $\wp$-function at $z=0$

Choosing the lattice $L=[1, \sqrt{-2}]$, what is the value of $\wp(0)$? The definition of the Weierstrass $\wp$ function for the lattice $L$ is $$ \wp(z;L) = \frac{1}{z^2} + \sum_{\omega \in L - \{ 0 \...
Iris's user avatar
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9 votes
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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} ...
User's user avatar
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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 $\...
Arvid Samuelsson's user avatar
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1 answer
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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 = ...
Chaitanya Tappu's user avatar
1 vote
1 answer
143 views

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 ...
Noa Arvidsson's user avatar
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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 ...
Abendsen's user avatar
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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 ...
Nico Schlömer's user avatar
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1 answer
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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 ...
user2284570's user avatar
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1 answer
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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 ...
Dqrksun's user avatar
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1 answer
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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}{\...
Dqrksun's user avatar
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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 ...
Dqrksun's user avatar
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2 votes
1 answer
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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), \...
user2554's user avatar
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8 votes
1 answer
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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 ...
Bumblebee's user avatar
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3 votes
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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 ...
D.R.'s user avatar
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2 votes
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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 ...
Mario's user avatar
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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 ...
André's user avatar
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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 ...
Bob Dobbs's user avatar
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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 ...
GeorgF's user avatar
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1 vote
0 answers
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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{...
Nomas2's user avatar
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1 vote
1 answer
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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{\...
emacs drives me nuts's user avatar
2 votes
1 answer
325 views

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}$...
User's user avatar
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3 votes
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}...
Nomas2's user avatar
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3 votes
2 answers
188 views

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 ...
Nomas2's user avatar
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0 answers
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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) =...
MarvinsSister's user avatar
1 vote
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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 ...
Kristóf Németh's user avatar
5 votes
1 answer
197 views

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 ...
Nomas2's user avatar
  • 667
9 votes
2 answers
436 views

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, ...
Nomas2's user avatar
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0 answers
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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 $A=0\to z=0$...
Bob Dobbs's user avatar
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4 votes
0 answers
162 views

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 ...
Artem Alexandrov's user avatar
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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}+\...
tianhaowu's user avatar
1 vote
1 answer
40 views

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 ...
japjap's user avatar
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0 answers
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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 ...
Math Attack's user avatar

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