# Questions tagged [elliptic-functions]

The elliptic-functions tag has no usage guidance.

88
questions

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### Is it true that all algebraic values of the $j$-invariant have a real Galois conjugate?

This is a follow-up of my recent question on real values of the $j$-invariant. It has had a partial response so far, establishing that any "reality root" $n$ is indeed rational. Now it ...

3
votes

1
answer

201
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### Arguments where the $j$-invariant has non-trivially real values — and a conjectured duality

The $j$-function (a.k.a. $j$-invariant up to a factor $1728$) is most often only considered for arguments where it yields real values.
It is well known that for $a,b,n\in\mathbb N$, $j\left(\dfrac {a+...

1
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0
answers

55
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### Taylor series of incomplete elliptic integral of first kind

Say I have an incomplete elliptic integral of first kind of the form
$$F(\varphi(z), k(z))=\int_0^{\varphi(z)} \frac{d \theta}{\sqrt{1-k(z)^2 \sin ^2 \theta}}$$
where each arguments are function of ...

1
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0
answers

103
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### How to prove this peculiar relationship between minimal polynomials of Ramanujan class invariants?

The Ramanujan class invariants (a.k.a. "Ramanujan-Weber class invariants")
are defined for $n>0$ by
$$G_n=2^{-1/4}e^{\pi\sqrt{n}/24}\prod_{k=0}^\infty \left(1+e^{-(2k+1)\pi\sqrt{n}}\right)...

7
votes

0
answers

173
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### Can an ellipse roll down a tilted sine curve without jumping?

Background
Assume that we have a solid ellipse with uniform density, and that it rolls along a curve.
In the following MO question, I asked along what curve an ellipse rolls down fastest. It was ...

4
votes

1
answer

166
views

### When is a solution $P(f'(x)) = Q(f(x))$ periodic or double periodic?

$\newcommand{\cl}{\operatorname{cl}}\newcommand{\sl}{\operatorname{sl}}\newcommand{\cm}{\operatorname{cm}}\newcommand{\sm}{\operatorname{cm}}$Consider the differential equation
$$P(f '(x)) = Q(f(x))$$
...

8
votes

1
answer

273
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### 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 ...

2
votes

0
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76
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### 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:
$$\vartheta_{00}(w) = \prod_{n = 1}^{\infty} (1-w^{2n})(1+w^{2n-1})^2,$$
$$\...

5
votes

0
answers

493
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### 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{...

0
votes

1
answer

193
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### 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 ...

3
votes

4
answers

478
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### 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....

0
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72
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### 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
...

1
vote

1
answer

145
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### 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|^...

3
votes

1
answer

255
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### 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

1
answer

314
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### 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>...

5
votes

0
answers

133
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### 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 ...

2
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289
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### 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\,\...

0
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0
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127
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### 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 ...

4
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1
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145
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### 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 ...

5
votes

1
answer

249
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### 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 $(1,1,1)$, $(1,-1,-1)$, $(-1,1,-1)$, $(-1,-1,1)$....

2
votes

1
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172
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### 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(...

0
votes

1
answer

493
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### 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 ...

3
votes

0
answers

158
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### 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 ...

0
votes

1
answer

164
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### 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 ...

1
vote

1
answer

266
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### 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 ...

2
votes

0
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127
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### 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 ...

5
votes

1
answer

256
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### 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}
...

0
votes

1
answer

284
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### 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(...

6
votes

1
answer

359
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### 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 ...

7
votes

1
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349
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### 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 ...

0
votes

0
answers

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### 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{...

3
votes

1
answer

485
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### 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{\...

8
votes

2
answers

620
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### 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 ...

3
votes

1
answer

310
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### 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{\...

0
votes

0
answers

356
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### 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 ...

2
votes

0
answers

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### 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:...

5
votes

2
answers

452
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### 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=\,-...

1
vote

1
answer

120
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### 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}...

1
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0
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122
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### 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 ...

0
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0
answers

79
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### 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, \...

10
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0
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### 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 ...

3
votes

1
answer

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### 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 $w=2\pi i z$, $...

3
votes

1
answer

337
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:
<...

10
votes

0
answers

344
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### 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}}+...

1
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0
answers

98
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### 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 ...

3
votes

1
answer

315
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### 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{...

1
vote

0
answers

85
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### 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 $...

5
votes

1
answer

340
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### 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_{...

3
votes

1
answer

1k
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### 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^...

1
vote

0
answers

94
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### 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 ...