Questions tagged [elliptic-functions]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
8 votes
1 answer
258 views

A real-valued analogue of the Weierstrass $\wp$ Function

I am interested in the following function: $$\mathcal{Q}(z) = \sum_{w \in L^*} \frac{1}{|z-w|^2} - \frac{1}{|w|^2} \, . $$ This function is analogous to the Weierstrass $\wp$ function, the only ...
Aobara's user avatar
  • 181
2 votes
0 answers
73 views

How to write the division values of $\operatorname{sn}(u;k)$ as rational functions of theta functions with zero argument?

Define the "thetanulls" (theta functions (https://dlmf.nist.gov/20) with one argument equal to zero) as follows: $$\vartheta_{00}(w) = \prod_{n = 1}^{\infty} (1-w^{2n})(1+w^{2n-1})^2,$$ $$\...
Nomas2's user avatar
  • 303
5 votes
0 answers
485 views

Is $\int \operatorname{sn}^2u\,\mathrm du$ really irreducible?

Let $\operatorname{sn}$, $\operatorname{cn}$, $\operatorname{dn}$ be Jacobian elliptic functions (https://dlmf.nist.gov/22). According to Greenhill, The integrals $$\operatorname{sn}^2u,\operatorname{...
Nomas2's user avatar
  • 303
0 votes
1 answer
176 views

Can we integrate arbitrary rational functions of Jacobian elliptic functions?

We can integrate arbitrary rational functions of the trigonometric functions because of the tangent half-angle substitution (https://en.m.wikipedia.org/wiki/Tangent_half-angle_substitution). This led ...
Nomas2's user avatar
  • 303
3 votes
4 answers
439 views

Asymptotic for Ramanujan's $\tau$-function

The Ramanujan's $\tau$-function is defined by $$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty \tau (n)q^n$$ where $|q|\lt 1$. Is there a known asymptotic formula for $\tau (n)$ or $|\tau (n)|$, i....
Nomas2's user avatar
  • 303
0 votes
0 answers
70 views

Minimal polynomial of $\operatorname{cd}\frac{4K}{n}$

Define $$K=\int_0^1 \frac{dx}{\sqrt{1-x^4}}$$ and the Jacobi elliptic function $\operatorname{cd}$ with modulus $i$ by $$\int_{\operatorname{cd}z}^1\dfrac{dx}{\sqrt{1-x^4}}=z$$ on $z\in [0,2K]$ and by ...
Nomas2's user avatar
  • 303
1 vote
1 answer
139 views

Decay estimates for simple elliptic equations

Let and let $p(|z|)$ be the radial solution of the following equation $$ \Delta p + 4q = 0\quad \text{in } \mathbb{R}^n $$ where $n\geq 2$, $0<\alpha<1$, $q \triangleq q(|z|) = \frac 1{1+ |z|^...
Davidi Cone's user avatar
3 votes
1 answer
249 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 $...
japjap's user avatar
  • 41
0 votes
1 answer
297 views

An identity for Weierstrass elliptic functions evaluation

Let $\wp(z), \zeta(z)$ and $\sigma(z)$ be the Weierstrass $\wp$, zeta and sigma functions associated to the ODE: $$\wp'(z)^2 = 4(\wp(z)-e_1)(\wp(z)-e_2)(\wp(z)-e_3)$$ and we assume $e_1=\frac{2-c}3>...
T. Amdeberhan's user avatar
5 votes
0 answers
126 views

Algebraic dependence of the elliptic functions

Let $\{f_i\}_{i=1}^{n}$ be $n$ elliptic functions in $\mathbb{C}$. We say that $f_1, \dots, f_n$ are algebraic dependent over $\mathbb{C}$ if there exists a polynomial of $n$ variables with constant ...
yaoxiao's user avatar
  • 1,654
2 votes
0 answers
284 views

Hecke operators acting on the Weierstrass $\wp$-function

Roughly speaking, my question is the following: Let $\wp(\tau, z)$ be the Weierstrass $\wp$-function, where $\tau \in \mathbf{H}$ and $z \in \mathbf{C}$. If $p$ is a prime number, can we define $T_p\,\...
Adithya Chakravarthy's user avatar
0 votes
0 answers
125 views

Exact value of $0<\wp^{-1}(2^{-{2/3}})<2$ with $\wp$ the Weierstrass elliptic function

I am investigating solutions to the differential equation $$\ddot{y}(t)=6y(t)^2,\dot{y}(0), y(0)=y_0>0.\tag{1}\label{1} $$ Let $\wp(t)$ be the Weierstrass elliptic function with elliptic invariants ...
Dispersion's user avatar
4 votes
1 answer
130 views

The origin and use of the term "equianharmonic" (elliptic function)

This question has been posted on History of Science and Mathematics stack exchange, but there was no answer or comments there. In Weierstrass notation, the principal elliptic function $\wp$ is a ...
Alexandre Eremenko's user avatar
5 votes
1 answer
229 views

What is the surface area of the finite part of the Cayley nodal cubic surface?

The Cayley nodal surface is defined by the equation $x^2+y^2+z^2-2xyz=1$. The finite part of the surface is the tetrahedral part bounded by the 4 nodes $(1,1,1)$, $(1,-1,-1)$, $(-1,1,-1)$, $(-1,-1,1)$....
LeechLattice's user avatar
  • 9,411
2 votes
1 answer
160 views

How can I calculate $\wp(αu), α\in \Bbb{C}$, $αL⊆L$

Let $\wp(u) = \frac{1}{u^2} + \sum\limits_{\omega \in L, \omega \neq 0} \left(\frac{1}{(u-\omega)^2} - \frac{1}{\omega^2}\right)$ be a Weierstrass pe function. My question is, how can I calculate $\wp(...
Duality's user avatar
  • 1,407
0 votes
1 answer
341 views

Approximation of Incomplete elliptic integral of first kind

How can we represent F(x,m) in the infinte polynominal of x,m? (Note that F(x,m) is the incomplete elliptical integral of the first kind, and I used its representation in the wikipedia) More ...
Joshuav's user avatar
  • 169
3 votes
0 answers
144 views

Is there a general relationship between definite integrals over functions involving the complete elliptic integral of the first kind and zeta values?

Background Let $\textbf{K}(k)$ be the complete elliptic integral of the first kind, where $k$ is its elliptic modulus [1]. Moreover, define $k' := \sqrt{1-k^{2}} $ as its complementary modulus. I've ...
Max Muller's user avatar
  • 4,435
0 votes
1 answer
139 views

Explicit solution of the Lamé equation for n=1

The Jacobi form of Lamé equation is given by \begin{equation} \left(\frac{d^2 }{du^2} - n(n+1)k^2 \operatorname{sn}^{2}(u)-E\right)\Psi (u) = 0, \end{equation} where $k\in(0, 1)$ is parameter ...
dannyt's user avatar
  • 51
1 vote
1 answer
255 views

Closed form roots for polynomial $x^9 + ax^6 + bx^5 + cx^3 + d = 0$

I know about Abel–Ruffini theorem, but I have a polynomial of special form. From "Beyond the Quartic Equation" by R.B. King (a very interesting book, btw) I've learned about Tschirnhaus ...
Moonwalker's user avatar
2 votes
0 answers
113 views

Elliptic functions

The Weierstrass $\wp$-function is given by $$ \wp(z,\tau)=\frac{1}{z^{2}}+\sum_{(k,l)\neq (0,0)}\left(\frac{1}{(z-(k\tau+l))^{2}}-\frac{1}{(k\tau+l)^{2}}\right). $$ Let $\lambda$ be primitive $n$th ...
Jack's user avatar
  • 397
5 votes
1 answer
235 views

Resources on the stationary Schrödinger equation with the soliton potential

I am studying the following Lamé equation in the Jacobi form \begin{equation} -\frac{d^2 v}{dx^2} - \left(2k^2 \operatorname{cn}^{2}\left [x\;|\;k\right ]\right)v = \lambda v, \end{equation} ...
dannyt's user avatar
  • 51
0 votes
1 answer
282 views

Can a doubly periodic function be locally univalent?

I am looking for a meromorphic doubly periodic function such that the function is locally univalent. A standard meromorphic doubly periodic funtion is the Weirestrass $\wp$ function, defined as $$\wp(...
student's user avatar
  • 1,320
6 votes
1 answer
350 views

How to work out this elliptic function?

Let $f(x) = \sum\limits_{(n,m)\in\mathbb{Z}^2} \frac{1}{(x+ n + i m )^2}$ If feel it should be $1/E(x)$ where $E$ is some elliptic function, like $sn^2$. But Wolfram Alpha is giving me some strange ...
zooby's user avatar
  • 255
7 votes
1 answer
324 views

What is the analogue of the Jacobi theta function in the Weyl representation?

It is known (see for example the associated Wikipedia entry) that the Jacobi theta function $$\vartheta(z; \tau) = \sum_{n\in\mathbb{Z}} \exp(\pi in^2\tau + 2\pi inz)$$ arises from a certain ...
Michael Barz's user avatar
0 votes
0 answers
19 views

Estimatives for elliptic systems involving the laplacian

Considering the problem \begin{equation} \left\{ \begin{array}[c]{11} \Delta(\Delta \chi -\chi) = 0 & \text{in } \Omega, \\ \Delta \chi -\chi = h_2 - h_1, & \text{on } \partial\Omega \\ \end{...
Bruno Mascaro's user avatar
3 votes
1 answer
450 views

Generating function of the product of Legendre polynomials

The generating function of the product of Legendre polynomials for the same $n$ is given by \begin{aligned} \sum_{n=0}^{\infty} z^{n} \mathrm{P}_{n}(\cos \alpha) \mathrm{P}_{n}(\cos \beta)&=\frac{\...
Kane's user avatar
  • 43
8 votes
2 answers
573 views

Can the theory of elliptic functions developed from purely geometric considerations?

I always had this question, but was unable to get a definitive answer to it. There is the theorem of division of the arc length of the lemniscate with ruler and compass. So I always wondered, is it ...
Kuga's user avatar
  • 181
3 votes
1 answer
284 views

Regarding the Weierstrass $\wp$-function of the hexagonal lattice

Playing with the Weierstrass $\wp$-function of the hexagonal (or triangular) lattice $\mathbb{T}$, $$ \wp'(z)^2 = 4 \wp(z)^3 - 1, $$ I noticed that the zeros of $\wp'(z) + \sqrt{3}$ are $$ \frac{\...
vassilis papanicolaou's user avatar
0 votes
0 answers
302 views

How to determine the closed form of this Fourier series?

Consider the Series $$ S(z) \equiv \sum_{n \in \mathbb{Z}, n \ne 0} \frac{ 1 }{ \sin n\pi \tau \sin 2n \pi \tau } e^{2\pi i n z} \ , \quad \operatorname{Im}\tau > 0 $$ I am trying to find its ...
Lelouch's user avatar
  • 857
2 votes
0 answers
294 views

Infinite sum of iterated integrals of matrix products

Originally asked over at Stackexchange (https://math.stackexchange.com/questions/4169812/infinite-sum-of-iterated-integrals-of-matrix-products), but this forum was deemed more appropriate. The problem:...
genus_3_amoeba's user avatar
4 votes
2 answers
427 views

About the integral $\int_{0}^{1}\frac{\log(x)}{\sqrt{1+x^{4}}}dx$ and elliptic functions

NOTE: I post this question on math.stackexchange but nobody answered, so I try here. For a work we need to evaluate the following integral $$\int_{0}^{1}\frac{\log\left(x\right)}{\sqrt{1+x^{4}}}dx=\,-...
Marco Cantarini's user avatar
1 vote
1 answer
119 views

Analytic function with q- difference equation involving theta

Consider the analytic space $\mathbb{C}^{*}$ with coordinate $z$. Let $q$ be some parameter with $|q|<1$ and define the analytic function $$\theta(z;q):=\sum_{n\in\mathbb{Z}}q^{\binom{n}{2}}(-z)^{n}...
user avatar
1 vote
0 answers
113 views

Algebraic relation amongst an elliptic function and its convolution

NOTE: I edited this question, following the comments of Alexander Eremenko and Paul Garrett. I have a question concerning elliptic functions that maybe you can help me shed light on. I am a ...
Stefano's user avatar
  • 105
0 votes
0 answers
79 views

The loss of double periodicity (ellipticity)

Consider a meromorphic function $f(\mathfrak{a}_1, \mathfrak{a}_2)$, such that $$ \begin{align} f(\mathfrak{a}_1, \mathfrak{a}_2) = f (\mathfrak{a}_1 + 1, \mathfrak{a}_2) = f(\mathfrak{a}_1 + \tau, \...
Lelouch's user avatar
  • 857
10 votes
0 answers
145 views

Is there some precise way in which modular cocycle $c/(c\tau +d)$ "is" the generator of $H^1(\mathcal{M}_{ell}, \omega^2) \cong \mathbb{Z}/12$?

It is semiclassical that the extension class $\text{Ext}^1(\omega^{-1},\omega)\cong H^1(\mathcal{M}_{ell},\omega^2)$ over the modular stack $\mathcal{M}_{ell}/\mathbb{Z}$ is nonvanishing, being cyclic ...
xir's user avatar
  • 1,964
3 votes
1 answer
193 views

Index and weight of Weierstrass $\sigma$-function as a Jacobi form, versus a statement in a note by Zagier

Let $$\sigma_L(w; \tau):=\frac{w}{\exp\left(\sum_{k\ge 2} 2G_k(q)\frac{w^k}{k!}\right)}$$ be the version of the Weierstrass $\sigma$-function which is used to orient $\text{tmf}$; here $w=2\pi i z$, $...
xir's user avatar
  • 1,964
3 votes
1 answer
311 views

Understanding the implementation of the $p$-adic(?) sigma function in SageMath

I'm trying to understand the (pretty undocumented) method .sigma() method for formal groups of elliptic curves, as listed here. The source code looks like this: <...
xir's user avatar
  • 1,964
10 votes
0 answers
337 views

Is this elliptic integral identity known?

Thinking about some physical problem, I came across the following identity: $$\phi^2\Pi\left(-\phi,\frac{1}{\sqrt{2}}\right)+\phi^{-2}\Pi\left(\phi^{-1},\frac{1}{\sqrt{2}}\right)=\frac{\pi}{\sqrt{2}}+...
Zurab Silagadze's user avatar
1 vote
0 answers
60 views

Second order ODE with Jacobi elliptic function coefficients

I posted this last week on the Mathematics Stack Exchange but have not been able to get an answer. I have read the rules and couldn't find anything against this, but please remove this question if ...
Ash's user avatar
  • 111
3 votes
1 answer
274 views

Conformal mapping between two right-angled triangles

I want to derive a conformal mapping $f\!:\!A\!\to\! B$ where $A=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq x \}$ and $B=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq \frac{...
niran90's user avatar
  • 167
1 vote
0 answers
84 views

Snoidal wave solutions of the $\phi^4$ model

I want to prove the existence of snoidal wave solutions of the $\phi^4$ model, given by $$u_{tt}-u_{xx}=u-u^3,\; (x,t) \in \mathbb{R}\times \mathbb{R}.$$ So, we are looking for solutions in the form $...
Guilherme's user avatar
  • 205
5 votes
1 answer
331 views

How to prove some identities about infinite product?

Recently, I read one paper titled Modular equations and approximations to π by Ramanujan, in which there are some formulas for $q=\pi i \tau$( where $\tau=x+yi, y>0$, hence $|q|<1)$ : $$\prod_{...
Jacob.Z.Lee's user avatar
3 votes
1 answer
1k views

Inverse of the incomplete elliptic integral of the second kind

The incomplete elliptic integral of the second kind $E(\varphi \, | \,k)$ is defined as follows: $$E(\varphi \, | \,k) = \int_0^\varphi \sqrt{1-k^2\sin^2\theta} \, \mathrm{d}\theta $$ Where $0<k^...
Descartes Before the Horse's user avatar
1 vote
0 answers
89 views

Construct a doubly periodic function on $\mathbb{C}$ using Jacobi elliptic functions with anti-holomorphic involution

I would like to find explicit examples of non-constant meromorphic doubly periodic $f(z)$ on $\mathbb{C}$, i.e. a meromorphic function on $\mathbb{C} / \Gamma$ for some lattice $\Gamma$, such that ...
Jack Moon's user avatar
5 votes
1 answer
427 views

Expressing the inverse Dixon function in terms of more familiar functions

If $x^3+y^3-3\alpha xy=1$, is there an expression for the integral $$\int_0^z \frac{\mathrm dx}{y^2-\alpha x}$$ in terms of more familiar functions? A.C. Dixon introduced the elliptic functions $\...
J. M. isn't a mathematician's user avatar
1 vote
1 answer
421 views

Weierstrass elliptic function in Laurent series form [closed]

Could anyone help me to figure out how $$ f_0(z) = \wp (\log z; i \pi, \log \rho) $$ where $\wp$ denotes the Weierstrass elliptic function and $i \pi$, $\log \rho$ are its half-...
Fareeda's user avatar
  • 45
16 votes
4 answers
2k views

Determination of special values of Eisenstein series

We have the Eisenstein series of weight $k$: $G_k(z)=\frac 1 2 \sum_{m,n} \frac 1 {(mz+n)^k}$. Can we evaluate it in closed form for some special values of $z$, eg. $z=i$ or $z=\omega$? It is clear by ...
FusRoDah's user avatar
  • 3,680
7 votes
1 answer
645 views

What are the modularity properties of Weierstrass sigma function?

I'm a little confused at the sigma orientation of tmf, see e.g. Witten genus and its references. The Weierstrass sigma function can be written as $$\sigma_L(z)(q)=\frac{z}{\exp\left(\sum_{k\ge 2} G_k(...
xir's user avatar
  • 1,964
7 votes
2 answers
440 views

Are these 5 the only eta quotients that parameterize $x^2+y^2 = 1$?

Given the Dedekind eta function $\eta(\tau)$, define, $$\alpha(\tau) =\frac{\sqrt2\,\eta(\tau)\,\eta^2(4\tau)}{\eta^3(2\tau)}$$ $$\beta(\tau) =\frac{\eta^2(\tau)\,\eta(4\tau)}{\eta^3(2\tau)}\quad\;$$...
Tito Piezas III's user avatar
3 votes
0 answers
99 views

Evaluating a Fermi gas problem for a SO(2N+1) matrix integral

I have the following multiple integral derived from a random matrix calculation I wish to evaluate $$\int_0^{\pi} dx_1 dx_2 \cdots dx_n \rho(x_1,x_2)\cdots \rho(x_n,x_1)$$ where the $\rho$ functions ...
Aran's user avatar
  • 181