3
votes
2answers
126 views

Elliptical Integrals

I was trying to figure out the length of the arc in a single cycle of a sinusoidal curve and I used the curve length formula to arrive at $$\int_0^{2\pi}\sqrt{1+\cos^2x}\ dx,$$ which I am fairly ...
2
votes
0answers
26 views

Expression of $2\int_{0}^{\frac{1}{r_{0}}}\frac{du}{\sqrt{\frac{r_{0}-r_{s}}{r_{0}^{3}}-u^{2}\left(1-u r_{s}\right)}}$ in terms of elliptic integrals

In Gravitation by Misner et al. 1973 the authors state that (calculus related to the Schwarzschild metric page 678) : ...
1
vote
0answers
57 views

Closed form of $\sum_{n=1}^\infty(-1)^{n+1} \frac xn \ln\left(1+\frac xn\right), \quad x \in (0,1)$

Is there a known closed form of the series below? $$\sum_{n=1}^\infty(-1)^{n+1} \frac xn \ln\left(1+\frac xn\right), \quad x \in (0,1]$$
8
votes
0answers
142 views

The closed form of $\sum_{n=0}^{\infty} \arcsin\left(\frac{1}{e^n}\right)$

In my study on some type of integrals I met the series below that I don't how to approach it. Of course, one of the obvious questions is: does it have a closed form? Before answering that, I need to ...
10
votes
5answers
290 views
1
vote
1answer
53 views

Infinite series representation of elliptic integrals $F(k,\phi)$ and $E(k,\phi).$

Here is my working so far. For a natural number $n,$$$2^{2n-1}\left(-1\right)^n\sin^{2n}\phi=\frac{\left(-1\right)^n}{2}\binom{2n}{n}+\sum\limits_{r=0}^{n-1}\binom{2n}{r}\left(-1\right)^r\cos ...
4
votes
4answers
233 views

Prove that these two curves have the same length

My midterms are approaching, and I was going through some of our past Calculus midterms when I stumbled upon this question from 1996: Show that these two curves, $$(\Gamma) : \frac ...
0
votes
1answer
61 views

How to get arc-length of polar function $r= 4(1-\sin{\phi})$?

How can I get arc-length of this polar function? $$ r= 4(1-\sin{\phi})$$ $$-\frac{\pi}{2}\leq\phi\leq\frac{\pi}{2}$$ I know that arc-length of polar function can get calculate by ...
0
votes
0answers
17 views

Relationship between elliptic integrals of first and second order.

A colleague and I are making an issue where order to reach the result we are asked to prove identity between elliptic integrals first and second order. This is: $$\frac{V}{\pi}\left( ...
12
votes
1answer
297 views

How to prove $\int_0^\pi\frac{\ln(2+\cos\phi)}{\sqrt{2+\cos\phi}}d\phi=\frac{\ln3}{\sqrt3}K\left(\sqrt{\frac23}\right)$?

How can I prove the following conjectured identity? $$\int_0^\pi\frac{\ln(2+\cos\phi)}{\sqrt{2+\cos\phi}}d\phi\stackrel?=\frac{\ln3}{\sqrt3}K\left(\sqrt{\frac23}\right),\tag1$$ where $K(x)$ is the ...
1
vote
0answers
50 views

Closed form double integral $ \int_{a}^{c}dr \int_{b}^{d} dr' \, \frac{r r'}{\sqrt{(r - a)(r' - b)(r-c)(r'-d)}} \frac{r_<^{\ell}}{r_>^{\ell+1}}$

Is there a closed form expression for $$ S_\ell = \int\limits_{a}^{c}dr \int\limits_{b}^{d} dr' \, \frac{r r'}{\sqrt{(r - a)(r' - b)(r-c)(r'-d)}} \frac{[\min( r , r')]^{\ell}}{[\max(r,r')]^{\ell+1}} ...
2
votes
0answers
76 views

Evaluation of $A$ in $2K(\sqrt{x}) = -\log(1 - x) + A + o(1)$ when $x \to 1^{-}$

Let $$K(k) = \int_{0}^{\pi/2}\frac{dx}{\sqrt{1 - k^{2}\sin^{2} x}}$$ be the complete elliptic integral of first kind where $0 < k < 1$. Let $k' = \sqrt{1 - k^{2}}$ be the complementary modulus. ...
5
votes
2answers
384 views

Closed form integral $\int_b^c \frac{x^2}{\sqrt{(x-a)(x-b)(c-x)(d-x)}} dx$

Is there a closed form expression for the definite integral $$I=\int_b^c \frac{x^2}{\sqrt{(x-a)(x-b)(c-x)(d-x)}} dx$$ for $a<b<c<d$? Mathematica 9.0 can do it for special cases using ...
2
votes
0answers
71 views

How to compute the arc length of $f(x) = ax + b \sin(x)$.

I would like to compute the length of the arc of $f(x) = a x + b \sin(x)$ (let's say from $0$ to $\alpha < 2\pi$.). The traditional method of computing it as the integral $\int_0^\alpha ...
25
votes
1answer
314 views

Conjectured closed form for $\int_0^1x^{2\,q-1}\,K(x)^2dx$ where $K(x)$ is the complete elliptic integral of the 1ˢᵗ kind

I am interested in a general closed-form formula for integrals of the following form: $$\mathcal{J}_q=\int_0^1x^{2\,q-1}\,K(x)^2dx,\tag0$$ where $K(x)$ is the complete elliptic integral of the 1ˢᵗ ...
20
votes
1answer
416 views

Simplify $\frac{_3F_2\left(\frac{1}{2},\frac{3}{4},\frac{5}{4};1,\frac{3}{2};\frac{3}{4}\right)}{\Pi\left(\frac{1}{4}\big|\frac{1}{\sqrt{3}}\right)}$

Is it possible to simplify the ratio $$\mathcal{E}=\frac{_3F_2\left(\frac{1}{2},\frac{3}{4},\frac{5}{4};\ 1,\frac{3}{2};\ \frac{3}{4}\right)}{\Pi\left(\frac{1}{4}\Big|\frac{1}{\sqrt{3}}\right)},$$ ...
2
votes
0answers
180 views

Separate incomplete elliptic integral into real and imaginary parts

I am working in a problem that involves Incomplete Elliptic Integrals of the First and Second kind of the form $F(\sin^{-1}x~|~m)$ and $E(\sin^{-1}x~|~m)$ where the parameters $m$, $x$ are real ...
3
votes
1answer
247 views

A tricky integral (flux of a point charge through a disk)

The integrals: $$ \oint \frac{r\,dr\,d\phi}{\left(L^2+r^2+h^2+2Lr\cos\phi\right)^{3/2}}\\ \oint \frac{dx\,dy}{\left((L+x)^2+y^2+h^2\right)^{3/2}} $$ If we have a point charge at the origin and we ...
3
votes
1answer
165 views

Identity concerning complete elliptic integrals

It can be easily checked that both the complete elliptic integrals $K(k), K'(k)$ satisfy the same second order differential equation $$kk'^{2}\frac{d^{2}y}{dk^{2}} + (1 - 3k^{2})\frac{dy}{dk} - ky = ...
7
votes
1answer
617 views

Derivative of the elliptic integral of the first kind

The complete elliptic integral of the first kind is defined as $$K(k)=\int_0^{\pi/2} \frac{dx}{\sqrt{1-k^2\sin^2{x}}}$$ and the complete elliptic integral of the second kind is defined as ...
4
votes
2answers
251 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
76 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 ...
3
votes
1answer
86 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 ...
1
vote
0answers
169 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 ...