Linked Questions

23
votes
3answers
795 views

Prove $\int_{0}^{\pi/2} x\csc^2(x)\arctan \left(\alpha \tan x\right)\, dx = \frac{\pi}{2}\left[\ln\frac{(1+\alpha)^{1+\alpha}}{\alpha^\alpha}\right]$

When I showed to my brother how I proved \begin{equation} \int_{0}^{\!\Large \frac{\pi}{2}} \ln \left(x^{2} + \ln^2\cos x\right) \, \mathrm{d}x=\pi\ln\ln2 \end{equation} using the following theorem by ...
29
votes
1answer
5k views

Fourier series of Log sine and Log cos

I saw the two identities $$ -\log(\sin(x))=\sum_{k=1}^\infty\frac{\cos(2kx)}{k}+\log(2) $$ and $$ -\log(\cos(x))=\sum_{k=1}^\infty(-1)^k\frac{\cos(2kx)}{k}+\log(2) $$ here: twist on classic log of ...
14
votes
3answers
436 views

Computing $\sum _{k=1}^{\infty } \frac{\Gamma \left(\frac{k}{2}+1\right)}{k^2 \Gamma \left(\frac{k}{2}+\frac{3}{2}\right)}$ in closed form

What tools other than beta function you might like to use here? $$\sum _{k=1}^{\infty } \frac{\displaystyle \Gamma \left(\frac{k}{2}+1\right)}{\displaystyle k^2 \Gamma \left(\frac{k}{2}+\frac{3}{2}\...
8
votes
8answers
407 views

Evaluate $\int_{0}^{\frac{\pi}{2}}\frac{x^2}{ \sin x}dx$

I want to evaluate $$\int_{0}^{\frac{\pi}{2}}\frac{x^2}{ \sin x}dx$$ First,I tried to evaluate like this: $$\int_{0}^{\frac{\pi}{2}}\frac{x^2}{ \sin x}dx=\int_{0}^{\frac{\pi}{2}}x^2\left(\frac{1+\cos ...
4
votes
4answers
579 views

Integral of $x\log(\sin x)$

This is from an old S level paper. I am struggling with part (ii). Any hints?
1
vote
5answers
213 views

integral $ \int_0^{\frac{\pi}{3}}\mathrm{ln}\left(\frac{\mathrm{sin}(x)}{\mathrm{sin}(x+\frac{\pi}{3})}\right)\ \mathrm{d}x$

We want to evaluate $ \displaystyle \int_0^{\frac{\pi}{3}}\mathrm{ln}\left(\frac{\mathrm{sin}(x)}{\mathrm{sin}(x+\frac{\pi}{3})}\right)\ \mathrm{d}x$. We tried contour integration which was not ...
2
votes
3answers
221 views

How do I solve this double definite integral?

$$ \int_{0}^{\pi}\int_{0}^{x/8}\ln\left(\,\sin\left(\,x - 8y\,\right)\,\right) \,\mathrm{d}y\,\mathrm{d}x $$ I am pretty sure the solution is $\displaystyle-\,\frac{\ln\left(\,2\,\right)\,\pi^{2}...
6
votes
2answers
98 views

On the integral $\int_0^{\frac{\pi}{2}}x^{2n+1}\cot(x)dx$

While investigating the function $$A(z)=\int_0^\frac{\pi}{2} \frac{\sin(zx)}{\sin(x)}dx$$ I stumbled upon the integral $$\int_0^{\frac{\pi}{2}}x^{2n+1}\cot(x)dx$$ when attempting to calculate the ...
2
votes
1answer
94 views

An integral of a cosine multiplied by a log of a cosine

My integration is rather rusty these days so I am keen to get any available help. I would like to find a general expression for the following integral (preferably in closed form, but a series ...
2
votes
1answer
34 views

Function with Fourier coefficient $1 / \lvert n \rvert$, $n \neq 0$? [on hold]

Is there a closed form formula for the periodic function $$f(x) = \sum_{n \in \mathbb{Z}\backslash \{0\}}\frac{1}{\lvert n \rvert} \mathrm{e}^{ \mathrm{i} n x}?$$
1
vote
0answers
64 views

Is there a converse to the Carleson-Hunter theorem?

The Carleson-Hunter theorem states that the Fourier series of a function $f\in L_p$ converges almost everywhere to $f$ if $p>1$. Suppose we know that a trigonometric series converges almost ...