Questions tagged [elliptic-functions]
Questions on doubly periodic functions on the complex plane such as Jacobi and Weierstrass elliptic functions.
538
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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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0
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48
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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
$$\wp＝\frac{1}{z^2}+\...
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1
vote
1
answer
28
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Transcendence of periods of the Weierstrass elliptic function
In the following, $\wp$ denotes the Weierstrass elliptic function (https://en.wikipedia.org/wiki/Weierstrass_elliptic_function), $g_2$ and $g_3$ are invariants and $\omega_1$ and $\omega_2$ are ...
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Inversion of the imcomplete elliptic integral of 2nd kind: $E(z|m)=x\;\Longleftrightarrow\; z=\text{bm}(x|m)$
I would like an opinion on this issue: similarly to the fact that
$$F(z|m)=x\Longleftrightarrow z=\text{am}(x|m)$$
I also tried to reverse the function $E(z|m)$ and got the following series (I ...
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0
answers
45
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Found the parameter of the elliptic integral of the first kind
Suppose I have:
$$\frac{p}{[K(p)-K(\frac{-\pi}{2},p)]^2} = x$$
$K(p)$ being the complete elliptic function of the first kind and $K(\theta,p)$ the incomplete elliptic function of the first kind.
How ...
1
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0
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33
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Elliptic function of the third kind $\Pi(n|m)$ for $n>1$
Can someone help me with the following problem?
I have the complete elliptic integral of the 3° kind defined as follow:
$$\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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1
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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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2
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0
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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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0
answers
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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
$$R(x)=\frac{1}{2\pi}\int_{\overline{...
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0
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Higher-order 'clover functions' are not elliptic--how to see this?
Define the '$n$-clover function' $ \phi _n $ as the inverse of the function
$$ f_n(r) := \int_0^r \frac{dx}{\sqrt{1-x^n}} $$
This is related to studying the arc length of clover curves (among other ...
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3
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1
answer
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Conjectured closed forms for Eisenstein-like series
This question is related to:
Eisenstein sum.
Being $q=e^{\pi}$, we have also:
\begin{align*}
\sum_{n=1}^{\infty}\frac{n(q^{n}(-1)^{n}+1)}{q^{2n}+2(-1)^{n}q^{n}+1}=-\frac{1}{24}\tag{1},
\end{align*}
\...
- 397
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Weierstrass' $\operatorname{Al}$ functions: imaginary transformation
Weierstrass defines his $\operatorname{Al}$ functions as follows:
We use the usual Glaisher notation for elliptic functions. Let
$$\operatorname{ns}(u,k)^2=\frac{1}{u^2}+1+\alpha_1 u^2+\cdots+\alpha_r ...
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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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1
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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 ...
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27
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2
answers
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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
$$\sum_{n=0}^\infty2^na_{n+1}\sqrt{a_n}=\frac{\Gamma^2(\frac14)-4\cdot\Gamma^2(\frac34)}{8\...
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0
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Confusion of Generator point in it's Montgomery Form and Weierstrass Form for secp256k1
I am using GEC Module (https://github.com/HareInWeed/gec) to perform point operations on secp256k1. Here, the generator point is defined as below
...
5
votes
2
answers
257
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Sum involving the Euler's Pentagonal Number function derivative.
Being:
$$\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},...
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1
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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 $...
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4
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3
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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(...
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Weierstrass factorization of $\operatorname{sn}$ and others
By the Weierstrass factorization theorem, we can write the elliptic function $\operatorname{sn}$ as
$$\operatorname{sn}u=\frac{B}{A}$$
where $B$ is a certain product over its zeros and $A$ is a ...
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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}...
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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.
(...
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When is a solution $P(f'(x)) = Q(f(x))$ periodic or double periodic?
Consider the differential equation
$$P(f '(x)) = Q(f(x))$$
Where $P(x),Q(x)$ are polynomials.
Examples are $f'(x) = 1 + f(x)^2$ where we get a tan solution and $f'(x)^2 = 4 f(x)^3 - g_2 f(x) - g_3$ ...
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Simplifying Elliptic Integrals 2 [closed]
According to wolfram, the following equation holds.
However, I do not understand the derivation process.
Could you please tell me how to derive it?
$$\int_0^1 \frac{u^4}{\sqrt{(1 - u^2) (1 - k u^2)}} ...
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0
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Finding a degree 1 elliptic function by relaxing conditions
Review: A doubly-periodic meromorphic function $f$, or an elliptic function, has a degree $d$ defined to be the sum of the orders of the poles in its fundamental parallelogram $\Gamma$. As $z$ ranges ...
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1
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Does the definition of the Weierstrass $\sigma$ function contain an extraneous term?
The Weierstrass $\sigma$ function of the lattice $\Lambda$ is defined by
$$\sigma (z;\Lambda)=z\prod_{\lambda\in\Lambda;\,\lambda\ne 0}\left(1-\frac{z}{\lambda}\right)\exp\left(\frac{z}{\lambda}+\frac{...
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Prove a partial differentiation equation w.r.t. $g_2,g_3$ for the Weierstrass sigma function
The Wolfram Functions Site
gives this partial differentiation equation w.r.t. $g_2,g_3$ for the Weierstrass sigma function
$$ z\frac{\partial \sigma(z; g_2, g_3)}{\partial z}
- 4g_2\frac{\partial \...
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2
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1
answer
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Finding an automorphism of the Riemann sphere that sends the branch points of the Weierstrass elliptic function $\wp$ to $(0, \infty, -1, 1)$
I’ve read on my lecture notes that we can find an automorphism of the Riemann sphere that sends the branch points of the Weierstrass elliptic function over the complex torus $X=\mathbb{C} / {\mathbb{Z}...
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1
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Zeroes and Poles of an Elliptic Function
Suppose that $a_1,...,a_r$ and $b_1,...,b_r$ are the zeroes and poles in the fundamental parallelogram of an elliptic function f.
Show that $$\sum_{n=1}^{r} a_n-b_n = n \omega_1 + m \omega_2$$ for ...
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Fourier series of a particular elliptic function
There is an established result that the Fourier expansion of a particular ratio of Jacobi elliptic theta functions:
$$\frac{\theta_1(x+y)\theta_1'(0)}{\theta_1(x)\theta_1(y)} = \cot(x)+\cot(y)+4\sum_{...
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2
answers
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Where does the Weierstrass expansion of $\operatorname{sn}$ come from?
In Table of Integrals, Series and Products (p. 869) by Gradshteyn and Ryzhik, I found the identity (called 'the Weierstrass expansion of $\operatorname{sn}$')
$$\operatorname{sn}u=\frac{B}{A}$$
where $...
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0
answers
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Jacobi elliptic function formula
Jacobi defined the elliptic function by
$u=\int_0^{\phi}\frac{d\theta}{\sqrt{1-k^2\sin^2(\theta)}}$
,$am(u)=\phi,sn(u)=\sin(am(u)),cn(u)=\cos(am(u))$
$dn(u)=\sqrt{1-k^2sn^2(u)},k'^2+k^2=1$
and he set
$...
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1
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0
answers
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Given two points $P$ and $Q$ on a fundamental parallelogram, construct an elliptic function with simple poles at $P, Q$ by contour integral of $\wp$
We know that every elliptic functions can be written as a rational function of $\wp$ and $\wp'$, where $\wp$ is the Weierstrass p function.. However, I am wondering how, given two arbitrary points $P$ ...
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1
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If two elliptic functions share the same poles and zeros (including multiciplity) then they are proportional
I’m trying to understand the following statement found on my lecture notes:
If two elliptic functions share the same poles and zeros (including
multiciplity) then they are proportional
I’m trying to ...
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11
votes
2
answers
291
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Integrals of $\int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} f(k)\text{d}k$
Consider a type of integrals
$$
\int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1}
f(k)\text{d}k
$$
where $K=K(k),K^\prime=K(\sqrt{1-k^2})$ are complete elliptic integrals, and $k$ is an elliptic ...
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2
votes
1
answer
40
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Using the first Liouville theorem
The first Liouville theorem states
Any elliptic function without poles is constant.
Let $\wp$ be the Weierstrass elliptic function of a lattice $L$ and let $\sigma$ be the Weierstrass sigma function ...
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1
vote
0
answers
54
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Show "hexagonal elliptic curves" have algebraic coefficients
Introduction:
Given any $\tau \in \mathbb H$, there is an elliptic curve $y^2=4x^3-g_2(\tau)x-g_3(\tau)$ which has $\{1,\tau\}$ as its lattice. Wikipedia gives an "explicit" relation between ...
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votes
1
answer
78
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How to manipulate differential equation into Weierstrass P form?
I have the diffeq $y'^2=\frac23 y^3+\alpha$ for some $\alpha\in\mathbb R$. This has two solutions in the form of a Weierstrass P elliptic function which are $$6^{1/3}\wp\left(\frac{x\pm C}{6^{1/3}};0, ...
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1
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0
answers
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Division of Bernoulli's lemniscate by ruler and compass
Bernoulli's lemniscate with unit half-width is given by the polar equation $r=\sqrt{\cos 2\theta}$. Using this curve, in the first quadrant, we define the function $\operatorname{arcsl}$ according to ...
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4
votes
1
answer
71
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Functional equation and elliptic functions
I'm sorry if this question is similar to a one already asked, I'm not aware of many basic facts about elliptic curves.
Let $g$ be a meromorphic function on $\mathbb{C}$, and $\Lambda$ be a lattice in $...
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votes
0
answers
37
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Proof that $\frac{1}{2\pi\operatorname{Im}(\tau)}\sqrt{\frac{J(\tau)}{J(\tau)-1}}$ is holomorphic for $\operatorname{Im}(\tau)\gt \frac{5}{4}$
Let $q=e^{2\pi i\tau}$ and
$$E_2(\tau)=1-24\sum_{n=1}^\infty n\frac{q^n}{1-q^n},$$
$$E_4(\tau)=1+240\sum_{n=1}^\infty n^3\frac{q^n}{1-q^n},$$
$$E_6(\tau)=1-504\sum_{n=1}^\infty n^5\frac{q^n}{1-q^n},$$
...
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votes
1
answer
73
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Understanding a proof of $\eta_1\omega_2-\eta_2\omega_1=2\pi i$
The Weierstrass zeta function $\zeta$ of a lattice $L$ is defined by
$$\zeta (z,L)=\frac{1}{z}+\sum_{\omega\in L\setminus\{0\}}\left(\frac{1}{z-\omega}+\frac{1}{\omega}+\frac{z}{\omega^2}\right)$$
and ...
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4
votes
1
answer
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Are these sources wrong about Jacobi elliptic functions, and can we fix the flaw?
I first noticed that three Youtube videos [1] [2] [3] give the same definitions of the Jacobi elliptic functions sn, cn, dn (see below), which are different from Wikipedia's. Then I noticed the links ...
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votes
2
answers
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Express $\sum_{n\in\mathbb{Z}} \left ( q^{(8n+1)^2}-q^{(8n+3)^2}\right )$
How can we find a expression for the following sum
$$
S=\sum_{n\in\mathbb{Z}}
\left ( q^{(8n+1)^2}-q^{(8n+3)^2}\right )
$$
where $q=e^{-\pi{K^\prime(k)}/{K(k)}}$
and $K(x)=\int_{0}^{1} \frac{1}{\sqrt{...
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votes
1
answer
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Deducing the relation $f(x)-f(x+1)+f(x+2)-\cdots = 0.5f(x)+Af'(x)+Bf''(x)+\cdots$ from Abel's "Studies on Elliptic Functions"
In Abel's famous article "Studies on Elliptic Functions", most of which I've understood, there is a formula that confused me.
It is like this:
$$f(x)-f(x+1)+f(x+2)-\cdots = 0.5f(x)+Af'(x)+Bf'...
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0
votes
1
answer
61
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If $f$ is a non-constant elliptic function, then $z\mapsto f'(z)/f(z)$ is a non-constant elliptic function
How can I prove
If $f$ is a non-constant elliptic function, then $z\mapsto f'(z)/f(z)$ is a non-constant elliptic function?
To prove that $z\mapsto f'(z)/f(z)$ is elliptic, it suffices to notice ...
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votes
0
answers
21
views
Arc length of a graph of an elliptic function
Exercise 2 on p. 190 of Greenhill's ''Applications of Elliptic Functions'' asks me to
rectify, by means of elliptic arcs, $y=\sin x$ and $y=\operatorname{sn}(x,k).$
For $y=\sin x$, I get
$$\int \...
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13
votes
1
answer
353
views
Hyperbolic sums $S(n,k)=\sum_{m=1}^{\infty} \frac{1}{\cosh(\pi m)^{n} \sinh(\pi m)^{k}}$
It can be verified that
$$
\sum_{n=1}^{\infty}\frac{1}{\cosh(\pi n)^3\sinh(\pi n)^2}
=\frac{11}{12}-\frac{3K}{2\pi}+\frac{K^2}{2\pi^2}-\frac{K^3}{\pi^3}$$
where $K=\frac{\Gamma\left ( \frac14 \right )^...
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votes
1
answer
34
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Implication of of uniformly elliptic matrices
I have been reading this paper and in the beginning of section 2 it is written something that I am failing to see why it holds. It is stated that if $A$ is a $dxd$ bounded, symmetric matrix such that ...
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