All Questions
Tagged with elliptic-integrals special-functions
7 questions
2
votes
2
answers
508
views
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
$$ \int_{b_1}^{b_2} \frac{db}{\...
5
votes
1
answer
234
views
Reduction of integral for geodesic area to elliptic integrals
In my paper on geodesics on an ellipsoid, I express the area
between a geodesic segment and the equator in terms of an indefinite
integral
$$\int
\frac{t(e'^2) - t(k^2\sin^2\sigma)}{e'^2-k^2\sin^2\...
- 181
5
votes
2
answers
2k
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 ...
- 26.9k
1
vote
2
answers
527
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 $...
- 11
0
votes
1
answer
830
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 ...
- 412
3
votes
1
answer
791
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" ...
- 123
13
votes
1
answer
1k
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^{...
- 13.9k