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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 ...
Nomas2's user avatar
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How to write the division values of $\operatorname{sn}(u;k)$ as rational functions of theta functions with zero argument?

Define the "thetanulls" (theta functions (https://dlmf.nist.gov/20) with one argument equal to zero) as follows: $$\vartheta_{00}(w) = \prod_{n = 1}^{\infty} (1-w^{2n})(1+w^{2n-1})^2,$$ $$\...
Nomas2's user avatar
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3 votes
0 answers
271 views

Derivation of an integral containing the complete elliptic integral of the first kind

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{...
r-nishi's user avatar
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3 votes
4 answers
535 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{...
user avatar
1 vote
0 answers
201 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 ...
0xbadf00d's user avatar
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5 votes
1 answer
248 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 $(1,1,1)$, $(1,-1,-1)$, $(-1,1,-1)$, $(-1,-1,1)$....
LeechLattice's user avatar
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0 votes
1 answer
430 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 ...
Joshuav's user avatar
  • 169
2 votes
2 answers
476 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}{\...
Filippo Caleca's user avatar
2 votes
0 answers
101 views

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 ...
Nomas's user avatar
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0 votes
1 answer
476 views

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 ...
Duality's user avatar
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6 votes
1 answer
590 views

Arithmetic-geometric mean for rationals?

Let $\operatorname{AGM}(x,y)$ be the arithmetic-geometric mean of $x$ and $y$. Given an error $\varepsilon>0$, a bound $b\in\mathbb R_+$ and a function $f:\mathbb R\rightarrow\mathbb R$ with $f(x)=...
VS.'s user avatar
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4 votes
2 answers
474 views

Kinds of differentials and algebraic groups

This Wikipedia article mentions that the analogues of differentials of the first/second/third kind for algebraic groups are abelian varieties/algebraic tori/linear algebraic groups. I guess ...
GTA's user avatar
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1 answer
385 views

Analytically continue complete elliptic integral over branch cut

Consider the function: $$f(a) = \frac{K\left(\frac{2 k(a) i}{g(a) + k(a) i}\right)}{\sqrt{g(a) + k(a) i}}$$ where $g(a)$ and $k(a)$ are smooth real-valued functions of a real parameter $a\in[0,1]$, ...
Greg Egan's user avatar
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5 votes
1 answer
214 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\...
cffk's user avatar
  • 181
2 votes
1 answer
527 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 ...
ina's user avatar
  • 23
5 votes
2 answers
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 ...
Stanley Yao Xiao's user avatar
11 votes
0 answers
136 views

Assymptotics of a Selberg type integral

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} \...
Hatem's user avatar
  • 111
1 vote
2 answers
507 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 $...
user48857's user avatar
1 vote
0 answers
167 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 ...
Walker's user avatar
  • 11
3 votes
1 answer
287 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 ...
integral2's user avatar
0 votes
1 answer
753 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 ...
Bryan Clair's user avatar
3 votes
1 answer
768 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" ...
Dong Wang's user avatar
  • 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^{...
Turbo's user avatar
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3 votes
2 answers
460 views

approximate equation involving elliptic integrals

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 ...
a guy on the street's user avatar