Questions tagged [elliptic-integrals]

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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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Separate complete elliptic integral of third kind into real and imaginary parts

I'm interested in the real and imaginary parts of the complete elliptic integral of the third kind $$\Pi(n,k)=\int_0^{\pi/2} \frac{d\theta}{(1-n\sin^2\theta)\sqrt{1-k^2\sin^2\theta}}$$ for $k>1$. ...
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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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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 ...
497 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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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" ...
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 ...