Questions on elliptic integrals, integrals that involve the square root of a cubic or quartic polynomial.
0
votes
1answer
29 views
How to compute complete elliptic integral of the first kind in explicit form using elementary functions?
How to compute complete elliptic integral of the first kind in explicit form using elementary functions?
If it is not possible to compute complete elliptic integral of the first kind in explicit way, ...
4
votes
2answers
93 views
How to solve this integral
Wolfram integrator is unable to solve this:
$$
\int_0^\pi \frac{1}{\sqrt {(a-\cos(t))^2+(b-\sin(t))^2+c^2}}\,\mathrm dt
$$
Any suggestions? Thanks!
10
votes
1answer
130 views
Closed form for $\int_1^\infty\int_0^1\frac{\mathrm dy\,\mathrm dx}{\sqrt{x^2-1}\sqrt{1-y^2}\sqrt{1-y^2+4\,x^2y^2}}$
Consider the following integral:
$$\mathcal{I}=\int_1^\infty\int_0^1\frac{\mathrm dy\,\mathrm dx}{\sqrt{x^2-1}\sqrt{1-y^2}\sqrt{1-y^2+4\,x^2y^2}}.$$
It can be represented as
...
1
vote
1answer
116 views
Incomplete elliptic integral of the second kind and the arc length of an ellipse - does a `simple` relation exist?
Short introduction
For a calculation I am working on I need to determine the arc length $l$ of a part of an ellipse in terms of the major axis $2a$, the minor axis $2b$ and the angle $\phi$.
I ...
2
votes
0answers
65 views
Calculating the integral $\int_{0}^{\pi /6}\sqrt{1-\left(\frac{R_s\sin \theta }{C_L}\right)^2} d\theta$
I want to integrate $I=\int\limits_{0}^{\pi /6}{\sqrt{1-{{\left( \frac{{{R}_{s}}\sin \theta }{{{C}_{L}}} \right)}^{2}}}d \theta}$.
I get incomplete elliptic integral $E(z\mid m)$ in the calculation by ...
1
vote
0answers
81 views
Approximations of the incomplete elliptic integral of the second kind
For a calculation I am working on I need to determine the arc length $l$ of a part of an ellipse in terms of the major axis $2a$, the minor axis $2b$ and the angle $\phi$. I know that this is a ...
4
votes
2answers
109 views
Help computing rational square root integral
Those integral techniques I thought I'd never need to remember are coming back to bite me. Is there a way to compute integrals of the form:
$$ \int_0^\infty \frac{1}{\sqrt{1 + a x^2 + b x^4}} dx $$
I ...
2
votes
1answer
55 views
An integral for the Earth's insolation
Consider the function
$$
[-\pi/2,\pi/2] \ni \theta \mapsto s_\beta(\theta) = \int_0^{2\pi}
\sqrt{ 1 - \left(\cos \theta \sin \beta \cos \gamma
- \sin \theta \cos \beta \right)^2} \, d \gamma
$$
for ...
1
vote
0answers
50 views
Complete Elliptic Integral of the 3rd Kind - Residual Computation
Let us consider the following function $f(a,k)$ in the interval $a,k\in (0,1]$ :
$$f(a,k)=\frac{2 \sqrt{1-a^2} \sqrt{a^2-k^2}}{\sqrt{a^2}}\Pi\left(a^2,k^2\right)$$
where $\Pi\left(a^2,k^2\right)$ is ...
2
votes
1answer
139 views
Circumference of the ellipse, elliptic integrals, and WolframAlpha
The arc length of a graph of a function $f(x)$ is
$$\mathcal{l} = \int_a^b \sqrt {1 + f'(x)^2} dx$$
If you choose $f$ to be a parametrization of the ellipse with $a=1, b=\frac{1}{2}$, i.e. $f(x) = ...
1
vote
1answer
99 views
How to show that the Complete Elliptic Integral of the First Kind increases in m?
How can you show that the complete elliptic integral of first kind
$
\displaystyle K(m)=\int_0^\frac{\pi}{2}\frac{\mathrm du}{\sqrt{1-m^2\sin^2 u}}$
that is the same as a series
$$K(m)=\frac{\pi}{2} ...
4
votes
0answers
116 views
Inversion of elliptic integral
I have an equation of the type
$$
p=\int_0^b\sqrt{\left(a^2-x^2\right)\left(b^2-x^2\right)}dx,
$$
in which $a$ and $b$ (with $a>b>0$) are (known) functions of some parameter $H$ (such that it is ...
3
votes
1answer
69 views
Can it possible to calculate this integral directly, probably concerning ellipti integral
A student asked me to help her calculating this problem: Assume the length $L$ of a curve is given, and the equation of the curve is
\begin{gather*}
y(x)=A \sin\Big(\pi x-\frac{\pi}{2}\Big), 0\leq ...
4
votes
2answers
286 views
Asymptotic approximation to incomplete elliptic integral of third kind at a pole - determine constant
while studying a physics problem I found that asymptotically the incomplete elliptic integral of the third kind, (using the Mathematica conventions, where it is called EllipticPi),
...
5
votes
2answers
150 views
How to prove $(x^2-1) \frac{d}{dx}(x \frac{dE(x)}{dx})=xE(x)$
$$E(x)=\int_0^{\frac{\pi}{2}} \sqrt{1-x^2 \sin^2 t}\, dt$$
Where $E(x)$ is complete elliptic integral of the second kind.
$u=\sin t$
$$E(x)=\int_0^{1} \frac{\sqrt{1-x^2 u^2}}{\sqrt{1-u^2}}\, du$$
...
1
vote
2answers
117 views
The integral relation between Perimeter of ellipse and Quarter of Perimeter
Ellipse Equation
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
$x=a\cos t$ ,$y=b\sin t$
$$L(\alpha)=\int_0^{\alpha}\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt$$
...
4
votes
1answer
164 views
Hypergeometric functions & integral
I'm having difficulty re-deriving a result a calculation from a paper. The integral is
$$\int_0^{2\pi} \int_0^{2\pi} ...
5
votes
1answer
150 views
Topology of Branch Cuts and Elliptic Integrals
In reading these notes (elliptic curves starting from elliptic integrals) I came across a couple claims about the topology of some complex surfaces.
On page 4, they discuss the integral
$$\phi(x) = ...
11
votes
2answers
240 views
Evaluating the elliptic integral $\int_{-\pi}^\pi\frac{dx}{\sqrt{(t-2\cos x)^2-4}}$
I have the following integral,
$$I(t)=\int_{-\pi}^\pi\frac{dx}{\sqrt{(t-2\cos x)^2-4}},$$
where $t>4$ is a real parameter. I know from messing around numerically and playing with Mathematica that
...
1
vote
0answers
104 views
Evaluating the average distance from a point in the unit disk to the disk
I am interested in finding the average Euclidean distance from a point $(x,y)\in\mathbb{D}_2$, the unit disk $\{(u,v):u^2+v^2\leq 1\}\subseteq\mathbb{R}^2$, to the disk $\mathbb{D}_2$. This amounts to ...
29
votes
2answers
1k views
Is this function decreasing on $(0,1)$?
While doing some research I got stuck trying to prove that the following function is decreasing
$$f(k):= k K(k) \sinh \left(\frac{\pi}{2} \frac{K(\sqrt{1-k^2})}{K(k)}\right)$$
for $k \in (0,1)$.
...
0
votes
2answers
158 views
complete elliptic integral of the first kind
I'm looking for any "closed form" for the coefficient of the $\ell$-th power of $K(x)$, the complete elliptic integral of the first kind.
Thanks.
0
votes
1answer
405 views
How to calculate the X Y coordinates of an ellipse with only the X and Y radius length?
I have an ellipse where the radius of x-axis = 100 and y-axis = 30.
I have 3 objects where I want to evenly distribute it along the ellipse.
I have already done this for a circle where both axis' ...
3
votes
2answers
162 views
Writing complete elliptic integral of first kind as a hypergeometric function
I am trying to show $K=\int_{0}^{\frac{\pi}{2}}\frac{dz}{\sqrt{1-k^{2}\sin^{2}(z)}}$ can be written as $\frac{\pi}{2} \mathstrut_{2}F_{1}(\frac{1}{2},\frac{1}{2};1;k^{2})$.
First I used ...
1
vote
1answer
224 views
Identity for complete elliptic integral of the second kind
Why does $|x-1| \; E\left(-\dfrac{4x}{(x-1)^2}\right) = |x+1| \; E\left(\dfrac{4x}{(x+1)^2}\right)$, where $E(m)$ is the complete elliptic integral of the second kind, with parameter $m$?
7
votes
0answers
225 views
Integral involving Complete Elliptic Integral of the First Kind K(k)
I have run into an integral involving the complete elliptic integral, which can be put into the following form after changing integration variables to the modulus:
...
13
votes
3answers
351 views
Can this integral $\int_0^{2\pi} \frac{d\theta}{(a^2 \cos^2 \theta +b^2\sin^2\theta)^{3/2}}$ be written in the form of a elliptic integral
I am trying to find the magnetic field due to an elliptic loop of wire. How to do integrals of the type $$\int_0^{2\pi} \frac{d\theta}{(a^2 \cos^2 \theta +b^2\sin^2\theta)^{3/2}}$$ Where a and b are ...
3
votes
0answers
126 views
Solid angle spanned by disc/rewriting expression with elliptic integrals
The solid angle spanned by a disc of unit radius, as observed from a point $(r,z)$ at a distance $z>0$ above a point in the disc plane with at distance $r>0$ to the center, can be expressed as ...
