# Questions tagged [elliptic-functions]

The elliptic-functions tag has no usage guidance.

**5**

votes

**2**answers

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

Given the Dedekind eta function $\eta(\tau)$, define,
$$\alpha(\tau) =\frac{\sqrt2\,\eta(\tau)\,\eta^2(4\tau)}{\eta^3(2\tau)}$$
$$\beta(\tau) =\frac{\eta^2(\tau)\,\eta(4\tau)}{\eta^3(2\tau)}\quad\;$$...

**3**

votes

**0**answers

84 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 ...

**0**

votes

**0**answers

42 views

### Rewriting an elliptic integral in terms of theta functions

I wish to demonstrate the following from arXiv:hep-th/9808043v2 (equations (3.2) to (3.4)). This is rewriting the following incomplete elliptic integral of the third kind
$ \Phi_{\tilde{h}}(h) = \...

**2**

votes

**1**answer

66 views

### Behaviour of elliptic functions near degenerate lattice

What can be said about elliptic functions (Weierstrass $\wp$, $\sigma$, Jacobi $\theta$, sn, cn, etc.) in the limit of degenerate lattice. By "degenerate" I mean $\tau = \omega_3/\omega_1$ tends to a ...

**6**

votes

**2**answers

360 views

### Solution of an equation with Jacobi theta function

I have been struggling with this equation for some time and I do not seem to find any conclusive answer (it's from my research, not a homework).
It has to do with the real solutions $x$ to the ...

**8**

votes

**2**answers

253 views

### Bounding an elliptic-type integral

Let $K>L>0$. I would like to find a good upper bound for the integral
$$\int_0^L \sqrt{x \left(1 + \frac{1}{K-x}\right)} \,dx.$$
An explicit expression for the antiderivative would have to ...

**2**

votes

**1**answer

78 views

### Limits of a quasiperiodic function with two pseudoperiods

Let $\beta$ be a real number such that $\beta^2\notin\mathbb{Q}$. For any smooth function $f$ on $\mathbb{R}$ that decreases sufficiently at infinity, for example a Gaussian function, let us define
$$
...

**-1**

votes

**1**answer

323 views

### Infinite sum and product associated with the Weierstrass elliptic function [closed]

Can anyone help me figure out how the identity below was obtained?
$ \frac{1}{\sqrt{(e_1-e_3)(e_2-e_3)}} = R \prod \limits_{n=1}^{\infty} \left(1 - \frac{1}{R^{4n}} \right)^{-4}\left(1 + \frac{1}{R^{...

**2**

votes

**0**answers

53 views

### Will a slightly differently shaped torus make this guess about plane sections of a torus true?

Jacobi's elliptic functions and plane sections of a torus
After Greg Egan posted an excellent answer to a question of mine, which I accepted, I posted my own answer, linked above. The question ...

**20**

votes

**5**answers

1k views

### The letter $\wp$; Name & origin?

Do you think the letter $\wp$ has a name? It may depend on community - the language, region, speciality, etc, so if you don't mind, please be specific about yours. (Mainly I'd like to know the English ...

**4**

votes

**2**answers

215 views

### Differentiating the inverse Weierstrass P-function

I will begin with some background:
The solutions $\theta$ of $$\cos \theta=x $$
constitute of two families, each of which is an arithmetic progression. Namely, if $\arccos x$ denotes any particular ...

**-2**

votes

**2**answers

358 views

### Expression for infinite product

can anyone show me how
$$\displaystyle\frac{4}{R}\displaystyle\Pi_{n=1}^{\infty} \left(\frac{1+R^{-4n}}{1+R^{-4n+2}}\right)^4= \frac{1}{R}\left(1+2 \sum_{n=1}^ {\infty} \frac{1}{R^{2n(n+1)}}\...

**3**

votes

**1**answer

333 views

### Jacobi and Weierstrass elliptic function

Jacobi elliptic function $\mathrm{sn}$ is defined as
$$\operatorname{sn}(u,k)=x\Leftrightarrow u=\int_0^x \frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}.$$
and Weierstrass sigma function $\sigma$ is defined as
...

**4**

votes

**2**answers

308 views

### Special values of the modular J invariant

A special value:
$$
J\big(i\sqrt{6}\;\big) = \frac{(14+9\sqrt{2}\;)^3\;(2-\sqrt{2}\;)}{4}
\tag{1}$$
I wrote $J(\tau) = j(\tau)/1728$.
How up-to-date is the Wikipedia listing of known special values ...

**8**

votes

**2**answers

749 views

### Jacobi's elliptic functions and plane sections of a torus

In $\mathbb R^3$ with Cartesian coordinates $(x,y,z),$ revolve the circle $(x-\sqrt 2)^2+z^2 =1,\ y=0$ about the $z$-axis.
This yields a torus embedded in $3$-space that is conformally equivalent to ...