Questions tagged [elliptic-functions]
Questions on doubly periodic functions on the complex plane such as Jacobi and Weierstrass elliptic functions.
538
questions
1
vote
0
answers
41
views
+100
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 ...
0
votes
0
answers
48
views
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
$$\wp=\frac{1}{z^2}+\...
1
vote
1
answer
28
views
Transcendence of periods of the Weierstrass elliptic function
In the following, $\wp$ denotes the Weierstrass elliptic function (https://en.wikipedia.org/wiki/Weierstrass_elliptic_function), $g_2$ and $g_3$ are invariants and $\omega_1$ and $\omega_2$ are ...
0
votes
0
answers
29
views
Inversion of the imcomplete elliptic integral of 2nd kind: $E(z|m)=x\;\Longleftrightarrow\; z=\text{bm}(x|m)$
I would like an opinion on this issue: similarly to the fact that
$$F(z|m)=x\Longleftrightarrow z=\text{am}(x|m)$$
I also tried to reverse the function $E(z|m)$ and got the following series (I ...
0
votes
0
answers
45
views
Found the parameter of the elliptic integral of the first kind
Suppose I have:
$$\frac{p}{[K(p)-K(\frac{-\pi}{2},p)]^2} = x$$
$K(p)$ being the complete elliptic function of the first kind and $K(\theta,p)$ the incomplete elliptic function of the first kind.
How ...
1
vote
0
answers
33
views
Elliptic function of the third kind $\Pi(n|m)$ for $n>1$
Can someone help me with the following problem?
I have the complete elliptic integral of the 3° kind defined as follow:
$$\Pi(n|m):=\int_0^{\frac{\pi}{2}}\frac{\mathrm{d}t}{(1-n\sin(t)^2)\sqrt{1-m\sin(...
1
vote
1
answer
70
views
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 $z\in\mathbb{C}$:
$$K(m)=F\left(\left.\frac{\pi}{2}-z\right|m\right)+\frac{1}{\sqrt{1-m}}\cdot F\left(z\left|\frac{m}{m-1}\right.\right)$$
$...
2
votes
0
answers
127
views
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(...
2
votes
0
answers
105
views
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
$$R(x)=\frac{1}{2\pi}\int_{\overline{...
0
votes
0
answers
32
views
Higher-order 'clover functions' are not elliptic--how to see this?
Define the '$n$-clover function' $ \phi _n $ as the inverse of the function
$$ f_n(r) := \int_0^r \frac{dx}{\sqrt{1-x^n}} $$
This is related to studying the arc length of clover curves (among other ...
3
votes
1
answer
255
views
Conjectured closed forms for Eisenstein-like series
This question is related to:
Eisenstein sum.
Being $q=e^{\pi}$, we have also:
\begin{align*}
\sum_{n=1}^{\infty}\frac{n(q^{n}(-1)^{n}+1)}{q^{2n}+2(-1)^{n}q^{n}+1}=-\frac{1}{24}\tag{1},
\end{align*}
\...
1
vote
0
answers
32
views
Weierstrass' $\operatorname{Al}$ functions: imaginary transformation
Weierstrass defines his $\operatorname{Al}$ functions as follows:
We use the usual Glaisher notation for elliptic functions. Let
$$\operatorname{ns}(u,k)^2=\frac{1}{u^2}+1+\alpha_1 u^2+\cdots+\alpha_r ...
0
votes
0
answers
59
views
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 ...
1
vote
0
answers
35
views
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)...
2
votes
1
answer
55
views
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 ...
27
votes
2
answers
1k
views
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
$$\sum_{n=0}^\infty2^na_{n+1}\sqrt{a_n}=\frac{\Gamma^2(\frac14)-4\cdot\Gamma^2(\frac34)}{8\...
0
votes
0
answers
34
views
Confusion of Generator point in it's Montgomery Form and Weierstrass Form for secp256k1
I am using GEC Module (https://github.com/HareInWeed/gec) to perform point operations on secp256k1. Here, the generator point is defined as below
...
5
votes
2
answers
257
views
Sum involving the Euler's Pentagonal Number function derivative.
Being:
$$\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
98
views
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
135
views
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(...
1
vote
0
answers
34
views
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 ...
2
votes
1
answer
194
views
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
votes
0
answers
92
views
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.
(...
1
vote
0
answers
37
views
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 $f'(x)^2 = 4 f(x)^3 - g_2 f(x) - g_3$ ...
-1
votes
1
answer
50
views
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?
$$\int_0^1 \frac{u^4}{\sqrt{(1 - u^2) (1 - k u^2)}} ...
0
votes
0
answers
32
views
Finding a degree 1 elliptic function by relaxing conditions
Review: A doubly-periodic meromorphic function $f$, or an elliptic function, has a degree $d$ defined to be the sum of the orders of the poles in its fundamental parallelogram $\Gamma$. As $z$ ranges ...
2
votes
1
answer
65
views
Does the definition of the Weierstrass $\sigma$ function contain an extraneous term?
The Weierstrass $\sigma$ function of the lattice $\Lambda$ is defined by
$$\sigma (z;\Lambda)=z\prod_{\lambda\in\Lambda;\,\lambda\ne 0}\left(1-\frac{z}{\lambda}\right)\exp\left(\frac{z}{\lambda}+\frac{...
1
vote
0
answers
47
views
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
$$ z\frac{\partial \sigma(z; g_2, g_3)}{\partial z}
- 4g_2\frac{\partial \...
2
votes
1
answer
51
views
Finding an automorphism of the Riemann sphere that sends the branch points of the Weierstrass elliptic function $\wp$ to $(0, \infty, -1, 1)$
I’ve read on my lecture notes that we can find an automorphism of the Riemann sphere that sends the branch points of the Weierstrass elliptic function over the complex torus $X=\mathbb{C} / {\mathbb{Z}...
0
votes
1
answer
73
views
Zeroes and Poles of an Elliptic Function
Suppose that $a_1,...,a_r$ and $b_1,...,b_r$ are the zeroes and poles in the fundamental parallelogram of an elliptic function f.
Show that $$\sum_{n=1}^{r} a_n-b_n = n \omega_1 + m \omega_2$$ for ...
4
votes
0
answers
93
views
Fourier series of a particular elliptic function
There is an established result that the Fourier expansion of a particular ratio of Jacobi elliptic theta functions:
$$\frac{\theta_1(x+y)\theta_1'(0)}{\theta_1(x)\theta_1(y)} = \cot(x)+\cot(y)+4\sum_{...
2
votes
2
answers
330
views
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 $...
0
votes
0
answers
65
views
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
$...
1
vote
0
answers
35
views
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 $P$ ...
3
votes
1
answer
86
views
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 ...
11
votes
2
answers
291
views
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 ...
2
votes
1
answer
40
views
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 ...
1
vote
0
answers
54
views
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 ...
2
votes
1
answer
78
views
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 $$6^{1/3}\wp\left(\frac{x\pm C}{6^{1/3}};0, ...
1
vote
0
answers
68
views
Division of Bernoulli's lemniscate by ruler and compass
Bernoulli's lemniscate with unit half-width is given by the polar equation $r=\sqrt{\cos 2\theta}$. Using this curve, in the first quadrant, we define the function $\operatorname{arcsl}$ according to ...
4
votes
1
answer
71
views
Functional equation and elliptic functions
I'm sorry if this question is similar to a one already asked, I'm not aware of many basic facts about elliptic curves.
Let $g$ be a meromorphic function on $\mathbb{C}$, and $\Lambda$ be a lattice in $...
0
votes
0
answers
37
views
Proof that $\frac{1}{2\pi\operatorname{Im}(\tau)}\sqrt{\frac{J(\tau)}{J(\tau)-1}}$ is holomorphic for $\operatorname{Im}(\tau)\gt \frac{5}{4}$
Let $q=e^{2\pi i\tau}$ and
$$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},$$
...
0
votes
1
answer
73
views
Understanding a proof of $\eta_1\omega_2-\eta_2\omega_1=2\pi i$
The Weierstrass zeta function $\zeta$ of a lattice $L$ is defined by
$$\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)$$
and ...
4
votes
1
answer
126
views
Are these sources wrong about Jacobi elliptic functions, and can we fix the flaw?
I first noticed that three Youtube videos [1] [2] [3] give the same definitions of the Jacobi elliptic functions sn, cn, dn (see below), which are different from Wikipedia's. Then I noticed the links ...
2
votes
2
answers
141
views
Express $\sum_{n\in\mathbb{Z}} \left ( q^{(8n+1)^2}-q^{(8n+3)^2}\right )$
How can we find a expression for the following sum
$$
S=\sum_{n\in\mathbb{Z}}
\left ( q^{(8n+1)^2}-q^{(8n+3)^2}\right )
$$
where $q=e^{-\pi{K^\prime(k)}/{K(k)}}$
and $K(x)=\int_{0}^{1} \frac{1}{\sqrt{...
-2
votes
1
answer
87
views
Deducing the relation $f(x)-f(x+1)+f(x+2)-\cdots = 0.5f(x)+Af'(x)+Bf''(x)+\cdots$ from Abel's "Studies on Elliptic Functions"
In Abel's famous article "Studies on Elliptic Functions", most of which I've understood, there is a formula that confused me.
It is like this:
$$f(x)-f(x+1)+f(x+2)-\cdots = 0.5f(x)+Af'(x)+Bf'...
0
votes
1
answer
61
views
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 ...
0
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
$$\int \...
13
votes
1
answer
353
views
Hyperbolic sums $S(n,k)=\sum_{m=1}^{\infty} \frac{1}{\cosh(\pi m)^{n} \sinh(\pi m)^{k}}$
It can be verified that
$$
\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 )^...
0
votes
1
answer
34
views
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 ...