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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Elliptic functions as inverses of Elliptic integrals

Let us begin with some (standard, I think) definitions. Def: An elliptic function is a doubly periodic meromorphic function on $\mathbb{C}$. Def: An elliptic integral is an integral of the form $$f(x) = \int_{a}^x R\left(t,\sqrt{P(t)}\right)\…
EuYu
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About the integral $\int_{0}^{1}\frac{\log(x)}{\sqrt{1+x^{4}}}dx$ and elliptic functions

For a work we need to evaluate the following integral $$\int_{0}^{1}\frac{\log\left(x\right)}{\sqrt{1+x^{4}}}dx=\,-_{3}F_{2}\left(\frac{1}{4},\frac{1}{4},\frac{1}{2};\frac{5}{4},\frac{5}{4};-1\right).\tag{1}$$ Classical approaches seem to lead…
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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, $$\begin{aligned} \int_0^1 K(k^{1/1})\,dk &= 2C\\…
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How to prove that $\frac{\eta^{14}(q^4)}{\eta^{4}(q^8)}=4\eta^4(q^2)\eta^2(q^4)\eta^4(q^8)+\eta^4(q)\eta^2(q^2)\eta^4(q^4)$?

How can we prove that $$\frac{\eta^{14}(q^4)}{\eta^{4}(q^8)}=4\eta^4(q^2)\eta^2(q^4)\eta^4(q^8)+\eta^4(q)\eta^2(q^2)\eta^4(q^4) \ ?$$ Here, $\eta(q)$ is the Dedekind Eta Function, which is defined by $$\eta(q)=q^{1/24}\prod_{n=1}^\infty…
Shobhit
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elliptic generalizations of Euler's trick

So Euler employed the following identity $$\sin(z) = z \prod_{n=1}^{\infty} \left[1-\left(\frac{z}{n\pi}\right)^{2}\right]$$ to evaluate $\zeta(2n)$, for $n\in\mathbb{N}$ I'm curious if there's been any research on an elliptic generalization of…
graveolensa
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How did Gauss sum Eisenstein series?

Entry 61 in Gauss's diary states (this is a translation from latin): From the integer powers of $$\int_0^1 \frac{dx}{\sqrt{1-x^4}}$$ depends $$\sum_{m,n}(\frac{m^4 - 6m^2n^2 +n^4}{(m^2+n^2)^4})^k$$. The expression inside the summation is:…
user2554
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Summation over Weierstrass $\wp$ functions

I've been trying to prove the following closed expression for a summation over Weierstrass $\wp$-functions: \begin{equation} \sum_{k=1}^{N-1} \wp_N(k) =…
Hrodelbert
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Weierstrass elliptic functions and ordinary differential equations

I am studying Elliptic functions for a University project with a particular focus on Weierstrass's theory. For the past few weeks I have been studying various basic properties of the $\wp$ function (the majority of the Elliptic functions section in…
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Torus and Elliptic curves

In a conference on elliptic curves (an introduction to the subject), the speaker said that an elliptic curve (I.e. an equation of the form $y^2=x^3+ax+b $ where the RHS has distinct roots) is, in the complex space, a torus/Riemann surface of genus…
Friedrich
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Curious integrals for Jacobi Theta Functions $\int_0^1 \vartheta_n(0,q)dq$

There are various identities for the Jacobi Theta Functions $\vartheta_n(z,q)$ on the MathWorld page and on the Wikipedia page. But I found no integral identities for these functions. Meanwhile, there are beautiful identities for the simple case of…
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Animation of Weierstrass $\wp$-function as a map from a torus to the sphere?

I am wondering if there exists somewhere an "animation" of one such map (for some lattice / torus), in the style of the kind of $z \mapsto z^2$ maps one encounters in complex analysis classes (one can somewhat convincingly visualize the plane…
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Global sections of vector bundles on a complex elliptic curve and analytic functions

Let me fix an elliptic curve $E$ over complex numbers with distinguished point $x \in E$. Thanks to Atiyah we know everything about discreet parameters of vector bundles and its moduli spaces. But I want to see more concrete description of sections.…
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Finite analog of these two infinite series with inverse tangents?

The identity $$ \sum_{n=1}^{\infty}\chi(n)\arctan e^{-\alpha n}+\sum_{n=1}^{\infty}\chi(n)\arctan e^{-\beta n}=\frac{\pi}{8}, \qquad \alpha\beta=\frac{\pi^2}{4},\tag{1} $$ where $\chi(n)=\sin\frac{\pi n}{2}$ is Dirichlet character modulo $4$, known…
Nemo
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Turning an elliptic curve over C into a complex torus

I have been reading a lot about the Weierstrass $\wp$ function and I understand the parameterization of an elliptic curve with the elliptic function( i.e. $x=\wp(z)$ and $y=\wp^\prime(z)$). I would like to parameterize my own elliptic curve, but I'm…
Brent J
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Questions about Weierstrass's elliptic functions

From the wikipedia: == In terms of the two periods, Weierstrass's elliptic function is an elliptic function with periods $\omega_1$ and $\omega_2$ defined as $$\wp(z;\omega_1,\omega_2)=\frac{1}{z^2}+ \sum_{n^2+m^2 \ne 0} \left\{…
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