All Questions
15
questions
22
votes
2
answers
576
views
A closed form for a triple integral involving Heron's formula
Let
$$S(x,y,z)=\frac14\sqrt{(x+y+z) (-x+y+z) (x-y+z) (x+y-z) }\tag1$$
(note that it's Heron's formula for the area of a triangle with sides of lengths $x,y,z$).
I'm trying to evaluate the following ...
- 47.3k
6
votes
2
answers
360
views
An integral related to Gamma value.
We have:
$$\int_{0}^{\pi/2}\frac{\sin{x}\log{(\tan{(x/2))}+x}}{\sqrt{\sin{x}}(\sin{x}+1)}dx=\pi-\frac{\sqrt{2\pi}\Gamma{(1/4)}^{2}}{16}-\frac{\sqrt{2}\pi^{5/2}}{2\Gamma{(1/4)}^{2}}\tag{1}.$$
As other ...
- 323
7
votes
0
answers
276
views
Evaluate $\int_{0}^{1} \frac{K(k)E(k)^2-\frac{\pi^3}{8} }{k} \text{d}k$ and $\int_{0}^{1} \frac{E(k)^3-\frac{\pi^3}{8} }{k} \text{d}k$
Let $K(k),E(k)$ be the complete elliptic integral of the first kind and second kind respectively, where $k$ is the elliptic modulus. Consider four integrals,
$$\begin{aligned}
&I_1=\int_{0}^{1} \...
- 5,370
7
votes
2
answers
499
views
Evaluate $\int_{0}^{\pi/2} \ln\left[ \tan\left ( \frac{\theta}{2}\right) \right ]^2 K\left ( \sin\theta \right )\text{d}\theta$
Let us define $K(x)$ as complete elliptic integral of the first kind, where $x$ is elliptic modulus. A possible closed-form is ($G$ denotes Catalan's constant.)
$$
\int_{0}^{\pi/2}
\ln\left[ \tan\left ...
- 5,370
8
votes
5
answers
611
views
Evaluation of $\int_0^1\frac{\log x\,dx}{\sqrt{x(1-x)(1-cx)}}$
Assume $c$ is a small real number.
QUESTION. What is the value of this integral in terms of the complete elliptic function $K(k)$?
$$\int_0^1\frac{\log x}{\sqrt{x(1-x)(1-cx)}}\,dx.$$
I got as far as ...
- 339
4
votes
1
answer
321
views
Prove that $\int_{0}^{1}x^2K^\prime(x)^3\text{d}x =\frac{\Gamma\left ( \frac14 \right )^8 }{640\pi^2} -\frac{\pi^4}{40}$
I need to prove the following result
$$
\int_{0}^{1}x^2K^\prime(x)^3\text{d}x
=\frac{\Gamma\left ( \frac14 \right )^8 }{640\pi^2}
-\frac{\pi^4}{40},
$$
where $K^\prime(x)=K\left(\sqrt{1-x^2}\right)$...
- 5,370
28
votes
3
answers
1k
views
Interesting closed form for $\int_0^{\frac{\pi}{2}}\frac{1}{\left(\frac{1}{3}+\sin^2{\theta}\right)^{\frac{1}{3}}}\;d\theta$
Some time ago I used a formal approach to derive the following identity:
$$\int_0^{\frac{\pi}{2}}\frac{1}{\left(\frac{1}{3}+\sin^2{\theta}\right)^{\frac{1}{3}}}\;d\theta=\frac{3^{\frac{1}{12}}\pi\...
- 3,343
19
votes
1
answer
876
views
Integral with arithmetic-geometric mean ${\large\int}_0^1\frac{x^z}{\operatorname{agm}(1,\,x)}dx$
The arithmetic-geometric mean$^{[1]}$$\!^{[2]}$ of positive numbers $a$ and $b$ is denoted $\operatorname{agm}(a,b)$ and defined as follows:
$$\text{Let}\quad a_0=a,\quad b_0=b,\quad a_{n+1}=\frac{...
- 47.3k
101
votes
1
answer
4k
views
Arithmetic-geometric mean of 3 numbers
The arithmetic-geometric mean$^{[1]}$$\!^{[2]}$ of 2 numbers $a$ and $b$ is denoted $\operatorname{AGM}(a,b)$ and defined as follows:
$$\text{Let}\quad a_0=a,\quad b_0=b,\quad a_{n+1}=\frac{a_n+b_n}2,...
- 47.3k
7
votes
2
answers
481
views
Find the ratio of integrals $\int_0^1 (1\pm x^4)^{-1/2}\,dx$
How to find this ratio
$$\frac{\displaystyle \int_{0}^{1}\frac{1}{\sqrt{1+x^{4}}}\mathrm{d}x}{\displaystyle \int_{0}^{1}\frac{1}{\sqrt{1-x^{4}}}\mathrm{d}x}$$
without evaluating each integral? The ...
- 5,281
15
votes
1
answer
558
views
Closed-forms of the integrals $\int_0^1 K(\sqrt{k})^2 \, dk$, $\int_0^1 E(\sqrt{k})^2 \, dk$ and $\int_0^1 K(\sqrt{k}) E(\sqrt{k}) \, dk$
Let denote $K$ and $E$ the complete elliptic integral of the first and second kind.
The integrand $K(\sqrt{k})$ and $E(\sqrt{k})$ has a closed-form antiderivative in term of $K(\sqrt{k})$ and $E(\...
- 12.4k
7
votes
1
answer
438
views
Closed form double integral $ \int_{a}^{c}dr \int_{b}^{d} dr' \, \frac{r r'}{\sqrt{(r - a)(r' - b)(r-c)(r'-d)}} \frac{r_<^{\ell}}{r_>^{\ell+1}}$
Is there a closed form expression for
$$
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}}
$$...
- 1,258
8
votes
2
answers
6k
views
Closed form integral $\int_b^c \frac{x^2}{\sqrt{(x-a)(x-b)(c-x)(d-x)}} dx$
Is there a closed form expression for the definite integral $$I=\int_b^c \frac{x^2}{\sqrt{(x-a)(x-b)(c-x)(d-x)}} dx$$ for $a<b<c<d$?
Mathematica 9.0 can do it for special cases using ...
- 1,258
28
votes
1
answer
924
views
Conjectured closed form for $\int_0^1x^{2\,q-1}\,K(x)^2dx$ where $K(x)$ is the complete elliptic integral of the 1ˢᵗ kind
I am interested in a general closed-form formula for integrals of the following form:
$$\mathcal{J}_q=\int_0^1x^{2\,q-1}\,K(x)^2dx,\tag0$$
where $K(x)$ is the complete elliptic integral of the 1ˢᵗ ...
- 47.3k
26
votes
1
answer
731
views
Simplify $\frac{_3F_2\left(\frac{1}{2},\frac{3}{4},\frac{5}{4};1,\frac{3}{2};\frac{3}{4}\right)}{\Pi\left(\frac{1}{4}\big|\frac{1}{\sqrt{3}}\right)}$
Is it possible to simplify the ratio
$$\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)},$$
...
- 13.2k