Questions tagged [elliptic-functions]

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

If $f:\mathbb{C} \to \mathbb{C}$ is elliptic, then there exists nonequal $t_0,t_1 \in \mathbb{C}$ such that $$ f(z) = f(z + t_0), f(z) = f(z + t_1) $$ the parallelogram $0,t_0,t_0+t_1,t_1$ is called the fundamental parallelogram of $f$. $f$'s value is entirely determined by its values on the fundamental parallelogram. By Liouville's theorem, any holomorphic elliptic function must be constant, so the usual elliptic functions are meromorphic. In fact, they must have at least two poles (with multiplicity) on the fundamental parallelogram.

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How to calculate an inner ellipse points that is always a set distance from an outer ellipse points

I have an Ellipse with known coordinates , I would like to know how I can create an inner ellipse coordinates that are exactly 5 inches perpendicular from the outer ellipse points. Please see the drawing at the link - I have the outer points x,y…
StixO
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Functional equations of $\lambda(\tau)$

We have the elliptic lambda function: $$\lambda(\tau)=\frac{e_3-e_2}{e_1-e_2}$$ We want to look at how $\lambda$ changes under a modular transformation: $$\omega'_2=a\omega_2+b\omega_1$$ $$\omega'_1=c\omega_2+d\omega_1$$ Now for $a$ and $d$ odd and…
George1811
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Why is Every Elliptic Function of Order $2$ the Möbius Tranform of a $\wp$-function?

I'm trying to prove that every elliptic function of order $2$ has the form $$f(z)=\frac{a\wp(z-z_0)+b}{c\wp(z-z_0)+d}$$ I've got the following so far. Let $f$ be an elliptic function of order 2. Then $f$ has $2$ poles and $2$ zeroes inside the…
Edward Hughes
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advice for curve fitting

I have numerically obtained some curves, corresponding with it I have also obtained some roots. I strongly believed these curves can be fitted with some (elliptic) functions taken the roots as parameters. However, due to the limitation of my…
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How to show an ODE system has no global solution

Starting from any $(x_0,y_0,z_0)\in \mathbb{C}^3$, can the following ODE system have a solution for all real number? \begin{align} x'(t) &=3 y^2(t) \\ y'(t) &=2 x(t) z(t)-1 \\ z'(t) &=0 \end{align} $$ x(0)=x_0, \quad y(0)=y_0, \quad z(0)=z_0 $$ I…
Koma
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How can I conclude this "gluing property" for these Sobolev functions?

Let $u \in W^{1,2}(\Omega) \cap C(\Omega)$, where $\Omega$ is an open bounded domain in $R^n$ with smooth boundary.Let $B(x,R) \subset \overline{B(x,R)} \subset \Omega $ a ball. Consider $u^{\star} \in W^{1,2}B(x,R)$ satisfying $\Delta u^{\star} =…
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Pole of elliptic function

Let $f:C→P1$ be such that $f(z+1)=f(z+i)=f(z)$ for all z∈C. Let $Γ=\{m+ni:m,n∈Z\}$. Show that if $f$ is holomorphic on $C∖Γ$, and $z⋅f(z)$ is bounded in a neighbourhood of $z=0$, then $f$ is constant. I am stuck in the part how a condition at zero…
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Generality of a solution of a nonlinear PDE

Given the nonlinear PDE $$ \partial^2\phi+\phi^3=0 $$ I consider the characteristics $\xi=\kappa\cdot x$. Then I look for a solution in the form $$ \phi(x)=a\cdot\chi(\xi). $$ Provided $\kappa^2=\frac{a}{2}$, I get the ODE defining of the…
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how to prove this identity (that Mathematica doesn't know) formula for the lemniscate constant?

How can we prove this identity? Which, btw, Mathematica know how to simplify so it is missing some fundamental identity( related to the lemniscate constant) $s$ \begin{equation} s = 2 \int_{- \pi}^{\pi} \left| \frac{\sin (t) - i}{(\sin (t) +…
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How to calculate $ \int_{0}^{2K(k)} dn^2(u,k)\;du$?

How to calculate $$ \int_{0}^{2K(k)} dn(u,k)^2\;du?$$ Where $dn$ is the Jacobi Elliptical function dnoidal and $k \in (0,1)$ is the modulus. I know from the Fórmula $(110.07)$ of [1] (see page 10) that $$ \int_{0}^{K(k)} dn(u,k)^2\;du=E(k),$$ where…
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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 lemniscate $r=\sqrt{\cos(2\theta)}$ (made a silly…
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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 * Rx + y * RadiusX * Ux)^2 + (x * RadiusY * Ry +…
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