# Questions tagged [elliptic-integrals]

The tag has no usage guidance.

22 questions
Filter by
Sorted by
Tagged with
206 views

I found the following formula in "INTEGRALS AND SERIES, vol.3" by Prudnikov, Brychkov and Marichev (page 188, eq.5). $$\int_0^{\infty} \frac{x^{\alpha-1}}{\sqrt{(a+x)^2+z^2}}K(\frac{2\sqrt{... 3 votes 4 answers 502 views ### Some Log integrals related to Gamma value Two years ago I evaluated some integrals related to \Gamma(1/4). First example:$$(1)\hspace{.2cm}\int_{0}^{1}\frac{\sqrt{x}\log{(1+\sqrt{1+x})}}{\sqrt{1-x^2}} dx=\pi-\frac{\sqrt {2}\pi^{5/2}+4\sqrt{... 1 vote
122 views

### Numerically compute the Schwarz-Christoffel mapping to the square

I want to map the upper-half plane $$\mathbb H:=\{z\in\mathbb C:\Im(z)>0\}$$ to $[0,1)^2$ by a conformal map. If I got this right, then such a mapping is given by the Schwarz-Christoffel mapping to ...
203 views

### What is the surface area of the finite part of the Cayley nodal cubic surface?

The Cayley nodal surface is defined by the equation $x^2+y^2+z^2-2xyz=1$. The finite part of the surface is the tetrahedral part bounded by the 4 nodes $(1,1,1)$, $(1,-1,-1)$, $(-1,1,-1)$, $(-1,-1,1)$....
213 views

### Approximation of Incomplete elliptic integral of first kind

How can we represent F(x,m) in the infinte polynominal of x,m? (Note that F(x,m) is the incomplete elliptical integral of the first kind, and I used its representation in the wikipedia) More ...
367 views

511 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 ...
1k 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 ...
Does any one know some references/ ideas on how to study the assymptotics as $N$ goes to $\infty$ of the following Selberg type integral $$\int _{\mathbb R^N} e^{-|x|^2}\ \prod_{1\le i<j\le N} \... 1 vote 2 answers 457 views ### Inversion of incomplete elliptic integral of third kind I would like to know whether there is any solution available on the inversion of elliptic integrals of the third kind (incomplete)? That means that given \Pi(n,u,m) = f(x), I would like to obtain ... 1 vote 0 answers 162 views ### Implementing boundary conditions to an ODE (involving elliptic integrals) I am trying to solve the following differential equation:$$ \frac{\mathrm{d} f}{\mathrm{d} x} = \frac{x^2-2 a}{\sqrt{4k^2-(x^2-2 a)^2}}, $$where a and k are constants (k is known and a is ... 3 votes 1 answer 274 views ### elliptic integral with singularities I need to calculate elliptic integrals with singularities, up to a huge number of digits (250-1000). The problem is that Wolfram Mathematica can't do so many digits, and Pari intnum doesn't handle ... 0 votes 1 answer 692 views ### Integrating the complete elliptic integral K I've run into the following integral: \int \frac{K(k)}{k} dk where K is the complete elliptic integral of the first kind K(k) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-k^2 \sin\theta}}. I've ... 3 votes 1 answer 730 views ### Are traditional notations for elliptic integrals/functions in Latin or Greek letters? I am doing some calculation involving elliptic integrals/functions, and find the notations confusing. In Wittaker-Watson, the "Jacobi's earlier notation" H(u) is called the Eta-function, so the "H" ... 13 votes 1 answer 978 views ### Connection between Infinite continued fractions, elliptic integrals and AGM It is known that at x=1, the following continued fraction represents \frac{4}{\pi} and can be approximated rapidly using Gauss' Arithmetic Geometric mean.$$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 ...