# Questions tagged [elliptic-integrals]

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15
questions

**3**

votes

**1**answer

321 views

### Arithmetic-geometric mean for rationals?

Let $\operatorname{AGM}(x,y)$ be the arithmetic-geometric mean of $x$ and $y$. Given an error $\varepsilon>0$, a bound $b\in\mathbb R_+$ and a function $f:\mathbb R\rightarrow\mathbb R$ with $f(x)=...

**2**

votes

**1**answer

144 views

### Kinds of differentials and algebraic groups

This Wikipedia article mentions that the analogues of differentials of the first/second/third kind for algebraic groups are abelian varieties/algebraic tori/linear algebraic groups. I guess ...

**0**

votes

**1**answer

150 views

### Analytically continue complete elliptic integral over branch cut

Consider the function:
$$f(a) = \frac{K\left(\frac{2 k(a) i}{g(a) + k(a) i}\right)}{\sqrt{g(a) + k(a) i}}$$
where $g(a)$ and $k(a)$ are smooth real-valued functions of a real parameter $a\in[0,1]$, ...

**7**

votes

**0**answers

174 views

### Reduction of a type of hyperelliptic integrals to elliptic integrals

(Was asked on Math.Stackexchange)
In [1] (hereinafter refered to as "the handbook"), it is said that
... the more general integral (Eq 575.16)
$$ \int R(\tau)\sqrt{(\tau-r_1)(\tau-r_2)(\tau-...

**4**

votes

**1**answer

172 views

### Reduction of integral for geodesic area to elliptic integrals

In my paper on geodesics on an ellipsoid, I express the area
between a geodesic segment and the equator in terms of an indefinite
integral
$$\int
\frac{t(e'^2) - t(k^2\sin^2\sigma)}{e'^2-k^2\sin^2\...

**2**

votes

**1**answer

406 views

### Ellipsoidal harmonics - A Series expansion for Lame functions of the second kind

$\underline{Intro \;to \;skip}$
In the theory of ellipsoidal harmonics, Lame functions of the second kind $F_n$ arise as the second linearly independent solution (the first being Lame functions of ...

**5**

votes

**2**answers

746 views

### Evaluating elliptic integrals

I am interested in evaluating some elliptic integrals, and I have not been able to secure a reference to do exactly what I need. Most of the references I've found seem to focus on reducing more ...

**11**

votes

**0**answers

126 views

### Assymptotics of a Selberg type integral

Does any one know some references/ ideas on how to study the assymptotics as $N$ goes to $\infty$ of the following Selberg type integral
$$\int _{\mathbb R^N} e^{-|x|^2}\ \prod_{1\le i<j\le N} \...

**1**

vote

**1**answer

249 views

### Inversion of incomplete elliptic integral of third kind

I would like to know whether there is any solution available on the inversion of elliptic integrals of the third kind (incomplete)?
That means that given $\Pi(n,u,m) = f(x)$, I would like to obtain $...

**1**

vote

**0**answers

132 views

### Implementing boundary conditions to an ODE (involving elliptic integrals)

I am trying to solve the following differential equation:
$$
\frac{\mathrm{d} f}{\mathrm{d} x} = \frac{x^2-2 a}{\sqrt{4k^2-(x^2-2 a)^2}},
$$
where $a$ and $k$ are constants ($k$ is known and $a$ is ...

**3**

votes

**1**answer

222 views

### elliptic integral with singularities

I need to calculate elliptic integrals with singularities, up to a huge number of digits (250-1000). The problem is that Wolfram Mathematica can't do so many digits, and Pari intnum doesn't handle ...

**0**

votes

**1**answer

469 views

### Integrating the complete elliptic integral K

I've run into the following integral:
$\int \frac{K(k)}{k} dk$
where $K$ is the complete elliptic integral of the first kind
$K(k) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-k^2 \sin\theta}}$.
I've ...

**3**

votes

**1**answer

471 views

### Are traditional notations for elliptic integrals/functions in Latin or Greek letters?

I am doing some calculation involving elliptic integrals/functions, and find the notations confusing.
In Wittaker-Watson, the "Jacobi's earlier notation" H(u) is called the Eta-function, so the "H" ...

**13**

votes

**1**answer

828 views

### Connection between Infinite continued fractions, elliptic integrals and AGM

It is known that at $x=1$, the following continued fraction represents $\frac{4}{\pi}$ and can be approximated rapidly using Gauss' Arithmetic Geometric mean.
$$C(x) = x + \frac{1^{2}}{2x + \frac{3^{...

**3**

votes

**2**answers

420 views

### approximate equation involving elliptic integrals

Dear Reader:
Let $K(k)$ and $E(k)$ be elliptic integrals of respectively the first and second kind, where $k$ is the elliptic modulus and $k'=\sqrt{1-k^2}$ is the complementary elliptic modulus.
I ...