Questions tagged [elliptic-functions]
Questions on doubly periodic functions on the complex plane such as Jacobi and Weierstrass elliptic functions.
5
votes
1answer
72 views
Addition formula for elliptic integral of second kind
Let $k\in(0,1)$ and the incomplete elliptic integral integral $E(u, k) $ be defined by $$E(u, k) =\int_{0}^{u}\operatorname {dn} ^2(t,k)\,dt\tag{1}$$ where $\operatorname {dn} (u, k) $ represents one ...
1
vote
1answer
35 views
Solving a 2nd order non-linear ODE
First off, please correct me if my title is wrong.
I want to solve an equation which has the following form:
$f'' + Af^3 + Bf = 0$
The closest I have gotten to such a form was when looking as the ...
0
votes
0answers
35 views
Jacobi sn: Does $w=\operatorname{sn}(z,m)$ send the line $\operatorname{Im}(z)=K'/2$ to the circle $\left \|w^4 \right \|=1/m?$
The picture below was generated in Mathematica and shows the image of a rectilinear grid in $\mathbb C$ under the elliptic mapping $\operatorname{sn}(z,m)$. The question here concerns the highlighted ...
0
votes
0answers
17 views
How to prove that a division of $\vartheta(z:\tau)$ functions has no poles
How do I prove that $f(z) + f(iz)$ has no poles? where $$f(z)=\dfrac{\vartheta^2(0;i)\vartheta^2(z+ \frac{1}{2};i)}{\vartheta^2(\frac{1}{2};i)\vartheta^2(z;i)}$$ and $$\vartheta(z;\tau)=\sum\limits_{n=...
1
vote
0answers
18 views
Jacobian elliptic function: unfamiliar definition
I am reading a physics book that uses a Jacobian elliptic function for a special purpose. For that i have to evaluate $\text{sn}(\zeta ; \gamma)$. However, both parameters are purely imaginary and the ...
1
vote
0answers
25 views
Weierstraß $\sigma$ function identity
Let $\Lambda$ be a lattice and $\sigma_{\Lambda_{\tau}}(z):= \sigma(z)= \prod_{w\in \Lambda\setminus \{0\}} \left(1 - \frac{z}{w}\right)\exp\left(\frac{z}{w}+\frac{z^2}{2w^2}\right) $ the Weierstraß ...
2
votes
0answers
57 views
Hyperelliptic function addition formula
$$x= \int_{0}^{f(x)} \frac{du}{\sqrt{1-u^2}\sqrt{1-k^2u^2}\sqrt{1-l^2u^2}}$$
$$f(0)=0$$
If we apply derivative operation for both sides, we get:
$$f'(x)=\sqrt{(1-f^2(x))(1-k^2f^2(x))(1-l^2f^2(x))}$$...
1
vote
5answers
79 views
$\sum_{\omega\in \mathbb{Z}(i)^*} |\omega|^{-2}$ does not converge.
I'm trying to prove that the series
$$
\sum_{\omega\in \mathbb{Z}(i)^*} |\omega|^{-2}
$$
The problem can be viewed as the sum above the fundamental region $\Omega^* = \{m\omega_1 + n\omega_2 : m,n\...
7
votes
2answers
138 views
Is it true that $\int_0^1 \big(K(k^{1/2})\big)^2\,dk = \frac{7}2\zeta(3)$?
Define the complete elliptic integral of the first kind as,
$$K(k) = \tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)$$
Part I. From the link above, we find some of the evaluations below,
...
8
votes
0answers
155 views
What is $\int_0^1 \left(\tfrac{\pi}2\,_2F_1\big(\tfrac13,\tfrac23,1,\,k^2\big)\right)^3 dk$?
As in this post, define the ff:
$$K(k)=K_2(k)={\tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)}$$
$$K_3(k)={\tfrac{\pi}{2}\,_2F_1\left(\tfrac13,\tfrac23,1,\,k^2\right)}$$
$$K_4(k)={\...
0
votes
0answers
46 views
Nonlinear ODE - possible transformations or analytical techniques?
I work with a problem that gives me a nonlinear 3rd order ODE. The equation in question is:
$f'''(x) - \alpha f(x) f''(x) + \alpha f'(x)^2 + k f'(x)=0$
The coefficient $k$ always turns out to be an ...
2
votes
0answers
68 views
Is this an elliptic curve $x^2+y^2=k^2(1+x^2y^2)$
Is this an elliptic curve $x^2+y^2=k^2(1+x^2 y^2)$ over rationals? If yes, how can we transform it into usual form? My friend sent me this curve.
13
votes
1answer
324 views
An intriguing pattern in Ramanujan's theory of elliptic functions that stops?
I. Define the ff integrals,
$$K(k)=K_2(k)=\int_0^{\pi/2}\frac{1}{\sqrt{1-k^2 \sin^2 x}}dx=\large{\tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)}$$
$$K_3(k)=\int_0^{\pi/2}\frac{\cos\left(...
5
votes
0answers
52 views
Zeroes of some degree of two elliptic functions
Let $\tau \in {\mathbb C}$ with $\mathrm{Im} \tau > 0$, $a,b \in {\mathbb Q}$ not both integers (it's not clear to me whether assuming only $a,b \in {\mathbb R}$ will make a difference to the ...
1
vote
0answers
66 views
About Sturm's bound
The next theorem is known as Sturm's bound.
Theorem:Let $\mathfrak{m}$ be a prime ideal in the ring of integers $\mathcal{O}$ of a number field $K$, and let $\Gamma$ be a congruence subgroup of of ...
7
votes
0answers
160 views
Finding a particular solution to a linear PDE
I want to solve the PDE
$$\frac{\partial u}{\partial t}+x_1(x_2-x_3) \frac{\partial u}{\partial x_1}+x_2(x_3-x_1) \frac{\partial u}{\partial x_2}+x_3(x_1-x_2) \frac{\partial u}{\partial x_3}=\sum_{i=1}...
2
votes
1answer
79 views
Showing that an elliptic function has no poles
Let $\Lambda = \{m \omega_1+n\omega_2; m,n \in \mathbb{Z}\}$ with $\omega_i \in \mathbb{C}$ with $\omega_2/\omega_1 \notin \mathbb{R}$ be a lattice. Define the Weierstrass $\mathscr{P}$ function on ...
0
votes
1answer
37 views
elliptic curve over GF(2^3)
This is $f(x) = x^3+x+3$ over $GF\left(2^3\right)$
How to know numbers of points on this equation?
How to find those points?
Is it an irreducible polynomial?
0
votes
1answer
32 views
Evaluating a derivative in gp/pari
Specifically, I'm trying to evaluate the derivative of the Weierstrass P function at a specific point.
I know that I can set up a function like the following
p(z) = ellwp([1,I],z);
Which will ...
1
vote
1answer
52 views
How to show that Jacobi sine function is doubly periodic
The Jacobi sine function can be defined using two definitions. The first is $\operatorname{sn}(u,m)=\sin(\phi)$ where $u=\int_0^{\phi}\frac{d\theta}{\sqrt{1-m\sin^2\theta}}$.
Alternativey it may be ...
-2
votes
1answer
61 views
Hektic Oscillator
It's fascinating that whereas the solution of the quartic oscillator (restoring force $\propto r^3$) problem can be expressed as the distance-from-origin of a point moving with unit speed along a ...
1
vote
1answer
40 views
Calculation of Elliptic Functions at Extreme Values of Eccentricity
The scenario of transverse oscillations of a mass in centre of an elastic string yields an oscillator with a restoring force that is purely cubic ... or a potential well that is quartic. The solution ...
0
votes
0answers
69 views
Questions related to the Riemann Xi function $\xi(s)$ and Jacobi theta functions $\vartheta_3(0,q)$
This question assumes the following definitions.
(1) $\quad\psi(x)=\sum\limits_{n=1}^\infty e^{-\pi\,n^2\,x}=\frac{1}{2} \left(\vartheta_3\left(0,e^{-\pi\,x}\right)-1\right)$
(2) $\quad f(x)=\sum\...
2
votes
0answers
49 views
Addition formula for $\text{sn}(u)=\text{sn}(u,k)$
I have just learned the definition of the first Jacobian elliptic function $\text{sn}(u)=\text{sn}(u,k)$, defined as the sine of the inverse function of
$$F(\phi,k):=\int_0^\phi \frac{dx}{\sqrt{1-k^2\...
0
votes
0answers
45 views
Location of the zeros of Dedekind Eta Function
Just a fast question, since I have not been able to find any answer for it online.
Where are the zeros of Dedekind eta function $\eta(s)$ located?
Apart from the trivial one as $s \to i \infty$, ...
3
votes
0answers
128 views
Show that by a translation poles on the boundary of an elliptic curve can be avoided.
In Complex Analysis by Freitag it is claimed that if there are poles on the boundary of an elliptic curve (the parallelogram) can be avoided by a-translation.
Is there any simple but rigorous proof ...
1
vote
0answers
20 views
Finding doubly periodic solutions to partial differential equations
Say I have a certain PDE in real variables $x$ and $y,$ which might be nonlinear, so that we can't necessarily just throw a Fourier series at it.
By way of some intuition, let's say, I have a very ...
0
votes
1answer
33 views
Is there a function with prescribed zeros and poles on an elliptic curve?
Let $T$ be the complex tore from the lattice $(1, \tau)$ where $im(\tau)>0$.
How to prove the existence of a meromorphic function on $T$, with divisor $(0) + (\frac{1}{2}) - 2 (\frac{\tau}{2})$ ? (...
4
votes
0answers
76 views
Exact value of Elliptic Integrals.
I was taking currently in a elementary calculus course where i found how to find arc lengths of a smooth continuous curve.
so here is how i started :
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\Rightarrow y=...
0
votes
1answer
40 views
Quotient of quarter periods K' and K of Jacobi elliptic functions
There are several ways to express the quarter period $K$,
$$
K(m)=\int_0^{\pi/2}\frac{\mathrm{d}\theta}{\sqrt{1-m\sin^2\theta}},
$$
as a power series (and thus for $K'=K(1-m)$ there are, too) and also ...
0
votes
0answers
37 views
Problem with the $\begin{eqnarray*} \wp(z) \end{eqnarray*}$ differential equation of Weierstrass
The Laurent series of $\wp(z)$ about $z=0$ is given by
$$
\wp(z) = z^{-2}+\sum_{k=1}^\infty (2k+1)G_{2k+2}z^{2k}
$$ and for $z\in\mathbb C$ and $z\notin \Lambda$ (=lattice) we have
$$
(\wp(z)')^...
0
votes
0answers
50 views
Reference Request - Elliptic Functions
From what I hear, elliptic functions were orginally introduced to find roots of polynomials, especially to help compute integrals. What is a book that emphasizes this motivation and approach to ...
0
votes
0answers
62 views
Correspondence between a complex torus and an elliptic curve
In The Arithmetic of elliptic Curves, on page 156, Silverman considers the map
$$\phi\colon\mathbb C/\Lambda\longrightarrow E\subset \mathbb P^2(\mathbb C)$$
$$z\longmapsto [\wp(z),\wp'(z),1]$$
where $...
1
vote
0answers
48 views
Unable to Understand the Abel's Theorem for Elliptic Functions
Definition: Let $f$ be an elliptic function on $\mathbb{C}$ and $\Omega \subset \mathbb{C}$ be the periodic lattice of $f$. The divisior of $f$ is defined as $$(f)=\sum_j n_jP_j$$
where $P_j$ are the ...
1
vote
0answers
71 views
Evaluation of complicated multiple integral
I have the following multiple integral I would wish to evaluate
$$\int_0^{2\pi} dx_1 dx_2 \cdots dx_n D(x_1,x_2)\cdots D(x_n,x_1)$$
where the $D$ functions are given by
$$ D(x,x') = \text{Sn}^2\...
0
votes
0answers
32 views
Explicit Values of the Jacobi Theta Function [duplicate]
The sum $\sum_{n=-\infty}^{\infty}\exp(-\pi n^2) $, or $\vartheta(0;i)$ (Jacobi Theta Function) has a closed form solution of $\frac{\pi^{\frac{1}{4}}}{\Gamma(\frac{3}{4})}$ but nowhere have I been ...
4
votes
1answer
59 views
$\wp(z)$ and points of order $3$
This problem comes from Koblitz's book Introduction to Elliptic Curves and Modular Forms page 41 problem 2 and it says
Let $$f_N(z) = N \Pi(\wp(z)-\wp(u))$$
Where the product is taken over nonzero $...
4
votes
1answer
47 views
The solutions of $(y')^2=P(y)$ with $\deg P \in \{3,4\}$.
Here is some background on my question. The Weierstraß function (as well as its translates) $\wp(x)$ solves the implicit first-order ODE
$$(y')^2=4y^3-g_2y-g_3.$$
Differentiating gives the second-...
1
vote
1answer
83 views
Converting a function of $\cos^2$ to a complete elliptic integral of the first kind
I am having a hard time following Equation 2.6 of Taib, Bachok Bin. "Boundary integral method applied to cavitation bubble dynamics." (1985). The equation is on the middle of page 8 of the document or ...
8
votes
4answers
267 views
How to derive relationship between Dedekind's $\eta$ function and $\Gamma(\frac{1}{4})$
I am trying to determine in what way to approach finding a connection between Dedekind's Eta Function, defined as $$\eta(\tau)=q^\frac{1}{24}\prod_{n=1}^\infty(1-q^n)$$
where $q=e^{2\pi i \tau}$ is ...
1
vote
1answer
35 views
$| \Re(\tau)| \leq 1/2$ for $\tau$ in fundamental region.
I've been working through Alfors section on elliptic functions and I'm stuck on an inequality in one of the proofs regarding the fundamental region (thereom 2 of 2.3 The canonical basis). We have the ...
2
votes
1answer
70 views
Properties of $\frac {\theta_{1}''(z|\tau)}{\theta_{1}(z|\tau)}$?
I'm looking for references concerning the properties of the function $\frac {\theta_{1}''(z|\tau)}{\theta_{1}(z|\tau)}$ where $\theta_{1}(z|\tau)$ is a Jacobi theta function defined here. I am trying ...
-1
votes
1answer
83 views
Find extreme points of a rotated ellipse equation on a given axis
I'm having hard time figuring out how to find the points where it is most extreme on the X and Y axis.
For example lets say I have an equation that describes an ellipse that is rotated:
(x * RadiusX ...
0
votes
1answer
35 views
Determine points on an eliptic curve when p is large? + Point at infinity clarification
I have come across this question here: Points on elliptic curve over finite field
Which describes how you can take every value $x$ from $0$ to $p-1$ and calculate the points of an elliptic curve.
...
2
votes
1answer
60 views
Calculating the lattice of the tori of a non-singular projective cubic curve
If $C$ is the curve in $\mathbb{C}\mathbb{P}^{2}$ defined by the zero set of the polynomial $P^{\lambda}(x,y,z) = y^{2}z - x(x-z)(x-\lambda z)$, for $\lambda$ not $0$ or $1$. Then we know that $C$ is ...
4
votes
2answers
215 views
How to show that cusp of congruence subgroup $\Gamma_0(4)$ is $0,\frac{1}{2},\infty$？
Let $\Gamma_0(4)$ be a congruence subgroup of $SL(2,\mathbb{Z})$ defined as
$$\Gamma_0(4)=\Big\{M=\begin{pmatrix}
a &b\\
c& d
\end{pmatrix}\in SL(2,\mathbb{Z}) | c \equiv 0\bmod 4\Big\}.$$
...
0
votes
1answer
50 views
Triangle Inequality: use to prove convergence (psi function elliptic functions)
The problem statement, all variables and given/known data
Attached
I understand the first bound but not the second.
I am fine with the rest of the derivation that follows after these bounds,
2. ...
0
votes
2answers
49 views
How can I draw a function of the nome $q$ on the unit disk?
This figure is the real part of the discriminant as a function of the nome $q$ on the unit disk. It is taken from the wiki link:
https://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions
I ...
1
vote
2answers
69 views
Finding zeroes and poles of the Weierstrass $\wp$ and $\wp^{'}$ function associated with a lattice.
I have a lattice, $\Omega$, with basis $\lbrace 2+i, 1+3i\rbrace$ and fundamental region (square) $P$ with vertices $1+2i,\ 2, \ -1+i\ $ and $-i$. I want to find the zeroes and poles of $\wp$ and $\wp^...
1
vote
0answers
58 views
Higher derivatives $\wp^{(2n)}$ of the Weierstrass $\wp$-function
I'm interested in considering the Weierstrass $\wp$-function denoted $\wp(\tau, z)$ as well as its higher derivatives $\wp^{(2n)}$, for all $n \geq 0$. The derivatives are with respect to the $z$ ...
