# Questions tagged [elliptic-integrals]

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### How did Ramanujan find $\sum_{n=0}^\infty (-1)^n\frac{(1/2)_n(1/4)_n(3/4)_n}{n!^3}\frac{644n+41}{25920^n}=\frac{288\sqrt{5}}{5\pi}?$

The formula $$\sum_{n=0}^\infty (-1)^n\frac{(1/2)_n(1/4)_n(3/4)_n}{n!^3}\frac{644n+41}{25920^n}=\frac{288\sqrt{5}}{5\pi}$$ (in older notation) appears as eq. 38 in Ramanujan's paper Modular equations ...
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### 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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### 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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### 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 ...
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### 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" ...
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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 ...