All Questions
25
questions
2
votes
0
answers
238
views
Showing $\int_{0}^{1}\frac{E(\tfrac{x}{\sqrt{x^2+8}})}{\sqrt{8-7x^2-x^4}}dx=\frac{1}{3}K(\frac{1}{\sqrt{2}})E(\frac{1}{\sqrt{2}})$
Context
$\begin{align}
K(k)=\int_{0}^{\pi/2}\frac{dt}{\sqrt{1-k^2\sin^2t}}\tag{1}
\end{align}$
and
$\begin{align}
E(k)=\int_{0}^{\pi/2}\sqrt{1-k^2\sin^2t}dt\tag{2}
\end{align}$
the complete elliptic ...
- 323
1
vote
1
answer
61
views
Closed form for $\int_0^{\pi/2}(\cos^2{\theta})^{1/n}d\theta$
Is there a closed form for $f(n)=\int_0^{\pi/2}(\cos^2{\theta})^{1/n}d\theta$ for every $n\in\mathbb{N}$ ?
I suspect there may be, because of the following apparent equalities.
$f(1)=\frac{\pi}{4}$
$f(...
- 25.8k
5
votes
3
answers
650
views
Alternative approaches to showing that $\Gamma'(1/2)=-\sqrt\pi\left(\gamma+\log(4)\right)$
Starting from the definition of the Gamma function as expressed by
$$\Gamma(z)=\int_0^\infty x^{z-1}e^{-x}\,dx\tag1$$
we can show that the derivative of $\Gamma(z)$ evaluated at $z=1/2$ is given by
$$\...
- 181k
4
votes
2
answers
394
views
On a log-gamma definite integral
A very famous log-gamma integral due to Raabe is
$$\int_0^1 \log \Gamma (x) \, dx = \frac{1}{2} \log (2\pi).$$
Several proofs of this result can be found here.
I would like to know about the ...
- 11.8k
5
votes
1
answer
126
views
Is there a way to simplify the solution to $\int_{1}^{e^{\frac{1}{e}}} x^{x^{x^{x^{...}}}} dx$
My result for this integral is as follows:
$$\int_{1}^{e^{\frac{1}{e}}} x^{x^{x^{....}}} = (e^{\frac{1}{e}})e - e - \frac{1}{2} - \sum_{k=1}^{\infty} \left( \frac{\gamma((k+2),(k))}{{k}^{(k+2)}\Gamma(...
6
votes
3
answers
322
views
Prove that $\int_0^1 \frac{x^2}{\sqrt{x^4+1}} \, dx=\frac{\sqrt{2}}{2}-\frac{\pi ^{3/2}}{\Gamma \left(\frac{1}{4}\right)^2}$
How to show
$$\int_0^1 \frac{x^2}{\sqrt{x^4+1}} \, dx=\frac{\sqrt{2}}{2}-\frac{\pi ^{3/2}}{\Gamma \left(\frac{1}{4}\right)^2}$$
I tried hypergeometric expansion, yielding $\, _2F_1\left(\frac{1}{2},\...
- 8,654
15
votes
3
answers
1k
views
Closed-form of log gamma integral $\int_0^z\ln\Gamma(t)~dt$ for $z =1,\frac12, \frac13, \frac14, \frac16,$ using Catalan's and Gieseking's constant?
We have the known,
$$I(z)=\int_0^z\ln\Gamma(t)~dt=\frac{z(1-z)}2+\frac z2\ln(2\pi)+z\ln\Gamma(z)-\ln G(z+1)$$
or alternatively,
$$I(z)=\int_0^z\ln\Gamma(t)~dt= \frac{z(1-z)}{2}+\frac{z}{2}\ln(2\pi) -(...
- 54.9k
4
votes
1
answer
171
views
closed form of the following integral :$\int_{0}^{\infty}- \sqrt{x}+ \sqrt{x}\coth (x) dx$?
I have tried to evaluate this:$\int_{0}^{\infty}- \sqrt{x}+ \sqrt{x}\coth (x)$ using the the following formula
$$2 \Gamma(a) \zeta(a) \left(1-\frac{1}{2^{a}} \right) = \int_{0}^{\infty}\Big( \frac{x^{...
- 7,195
2
votes
1
answer
130
views
On a closed form for $\int_{-\infty}^\infty\frac{dx}{\left(1+x^2\right)^p}$ [duplicate]
Consider the following function of a real variable $p$ , defined for $p>\frac{1}{2}$:
$$I(p) = \int_{-\infty}^\infty\frac{dx}{\left(1+x^2\right)^p}$$
Playing around in Wolfram Alpha, I have ...
- 3,309
3
votes
1
answer
284
views
Definite Gamma function integral
Curiosity Question
It's very well known that
$$\int_a^{a+1} \ln\Gamma(x)\ dx = \frac{1}{2}\ln(2\pi) - a - \ln(a) - (a+1)\ln\Gamma(a) + (a+1)\ln\Gamma(1+a)$$
Clearly provided that $\Gamma(a)\geq ...
- 26.3k
3
votes
2
answers
177
views
Evaluate $\int_0^{\pi/2}\frac{1-\sqrt[18]{\cos u}}{1-\cos u}du$, in terms of particular values of special functions and constants
I wondered about this question when I've considered similar integrals with different integrand as an analogous of a formula for harmonic number.
As example that I know that can be calculated using ...
user243301
1
vote
2
answers
345
views
Closed form solution for $\int_{0}^{2\pi}|\cos^N x||\sin^M x|\cos^n x\sin^m x dx$
I am trying to calculate Fourier series coefficients (by hand) and the integrals I need to solve are of the following type
$$I(N,M,n,m)=\int_{0}^{2\pi}|\cos^N x||\sin^M x|\cos^n x\sin^m x dx,$$
in ...
- 15.9k
14
votes
1
answer
1k
views
Prove $\int_0^1 \frac{4\cos^{-1}x}{\sqrt{2x-x^2}}\,dx=\frac{8}{9\sqrt{\pi}}\left(9\Gamma(3/4)^2{}_4F_3(\cdots)+\Gamma(5/4)^2{}_4F_3(\cdots)\right)$
Mathematica gives the following. But how?!
$$\small{\int_0^1 \dfrac{4\cos^{-1}x}{\sqrt{2x-x^2}}\,dx=\frac{8}{9\sqrt{\pi}}\left(
9\Gamma\left(\tfrac{3}{4}\right)^2{}_4F_3\left( \begin{array}{c}\...
- 27k
9
votes
1
answer
448
views
Simple closed form for $\int_0^\infty\frac{1}{\sqrt{x^2+x}\sqrt[4]{8x^2+8x+1}}\;dx$
Some time ago, I used a fairly formal method (in the second sense of this answer) to derive the following integral, and am wondering whether it is correct or not:
$$\int_0^\infty\frac{1}{\sqrt{x^2+...
- 3,343
13
votes
2
answers
368
views
Prove known closed form for $\int_0^\infty e^{-x}I_0\left(\frac{x}{3}\right)^3\;dx$
I know that the following identity is correct, but I would love to see a derivation:
$$\int_0^\infty e^{-x}I_0\left(\frac{x}{3}\right)^3\;dx=\frac{\sqrt{6}}{32\pi^3}\Gamma\left(\frac{1}{24}\right)\...
- 3,343