# Questions tagged [elliptic-integrals]

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### 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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### 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 ...
Does any one know some references/ ideas on how to study the assymptotics as $N$ goes to $\infty$ of the following Selberg type integral $$\int _{\mathbb R^N} e^{-|x|^2}\ \prod_{1\le i<j\le N} \... 1 vote 1 answer 354 views ### Inversion of incomplete elliptic integral of third kind I would like to know whether there is any solution available on the inversion of elliptic integrals of the third kind (incomplete)? That means that given \Pi(n,u,m) = f(x), I would like to obtain ... 1 vote 0 answers 150 views ### Implementing boundary conditions to an ODE (involving elliptic integrals) I am trying to solve the following differential equation:$$ \frac{\mathrm{d} f}{\mathrm{d} x} = \frac{x^2-2 a}{\sqrt{4k^2-(x^2-2 a)^2}}, $$where a and k are constants (k is known and a is ... 3 votes 1 answer 247 views ### elliptic integral with singularities I need to calculate elliptic integrals with singularities, up to a huge number of digits (250-1000). The problem is that Wolfram Mathematica can't do so many digits, and Pari intnum doesn't handle ... 0 votes 1 answer 559 views ### Integrating the complete elliptic integral K I've run into the following integral: \int \frac{K(k)}{k} dk where K is the complete elliptic integral of the first kind K(k) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-k^2 \sin\theta}}. I've ... 3 votes 1 answer 638 views ### Are traditional notations for elliptic integrals/functions in Latin or Greek letters? I am doing some calculation involving elliptic integrals/functions, and find the notations confusing. In Wittaker-Watson, the "Jacobi's earlier notation" H(u) is called the Eta-function, so the "H" ... 13 votes 1 answer 938 views ### Connection between Infinite continued fractions, elliptic integrals and AGM It is known that at x=1, the following continued fraction represents \frac{4}{\pi} and can be approximated rapidly using Gauss' Arithmetic Geometric mean.$$C(x) = x + \frac{1^{2}}{2x + \frac{3^{...
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 ...