Questions tagged [elliptic-integrals]

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Definite integral of the square root of a polynomial ratio

I found myself with the following integral $$ \int_{b_1}^{b_2} \sqrt{\frac{(b-b_1)(b_2-b)(b_3-b)}{(b_4-b)}} \ db $$ with $ b_1 < b_2 < b_3 < b_4 $. I know that $$ \int_{b_1}^{b_2} \frac{db}{\...
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2 votes
0 answers
83 views

Determine whether $\frac{K\left(\sqrt{1-x^2}\right)^2}{K(x)^2}$ is positive rational (given $x$)

Denote the complete elliptic integral of the first kind by $$K(x)=\int_0^{\pi /2}\frac{d\varphi}{\sqrt{1-x^2\sin^2\varphi}}$$ and $$f(x)=\frac{K\left(\sqrt{1-x^2}\right)^2}{K(x)^2}$$ Question: Given a ...
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1 vote
1 answer
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To integrate elliptic integral, we glue two Riemann surface to make torus

To deal with elliptic integral, we often cut riemann surface and glue them together, and gain a torus. We do this in order to avoid indeterminacy of integral, in other word, to avoid the condition ...
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  • 321
6 votes
1 answer
578 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)=...
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4 votes
2 answers
408 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 ...
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0 votes
1 answer
272 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 $a\in[0,1]$, ...
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5 votes
1 answer
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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\...
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  • 181
2 votes
1 answer
478 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 ...
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5 votes
2 answers
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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 ...
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11 votes
0 answers
130 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} \...
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1 vote
1 answer
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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 $...
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1 vote
0 answers
150 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 ...
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3 votes
1 answer
247 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 ...
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0 votes
1 answer
559 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 ...
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3 votes
1 answer
638 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" ...
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13 votes
1 answer
938 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^{...
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  • 13k
3 votes
2 answers
427 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 ...
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