All Questions
8 questions
106
votes
5
answers
10k
views
integral of a "sin-omial" coefficients=binomial
I find the following averaged-integral amusing and intriguing, to say the least. Is there any proof?
For any pair of integers $n\geq k\geq0$, we have
$$\frac1{\pi}\int_0^{\pi}\frac{\sin^n(x)}{\...
- 43.2k
9
votes
1
answer
458
views
Summing moments and Riemann zeta values
Let $d\mu_n(x)=\cos^{2n}x\,dx$ and consider the averages of moments
$$\alpha_n=\frac{\int_0^{\pi/2}x^4d\mu_n(x)}{\int_0^{\pi/2}d\mu_n(x)}.$$
Then, I have encountered a curious evaluation
$$\sum_{n=1}^{...
- 43.2k
6
votes
0
answers
431
views
How to prove these identities for $\log(2)$ based on $_3F_2$ integrals?
In this MO post I have placed 4 Ramanujan-type hypergeometric series found using the LLL algorithm for fast computing of some logarithms. I could prove 3 of them by means of classical methods based on ...
- 2,836
4
votes
1
answer
143
views
Mean value of a function associated with continued fractions
Suppose that an irrational $x$ in $(0,1)$ has convergents $c(k,x)$, and let
$$d(x) = \sum_{k=0}^{\infty} \mid x - c(k,x)\mid.$$
What is the mean value of $d$?
- 3,443
3
votes
2
answers
303
views
Basic question related to Stieltjes integral
I am reading this paper. I am stuck on something, which I think is something basic but I haven't been able to figure it out yet, and I was hoping someone could explain it to me.
Let
$$
\sigma(u) = \...
- 3,625
2
votes
0
answers
249
views
Calculating $\int_1^{\infty}\frac{\operatorname{ali}(x)}{x^3}dx$, where $\operatorname{ali}(x)$ is the inverse function of the logarithmic integral
It is well-known that we can compute the closed-form of the integrals $$\int_1^{\infty}\frac{\log x}{x^2}dx$$
and $$\int_1^{\infty}\frac{\operatorname{li} (x)}{x^3}dx,$$
where $\operatorname{li} (x)$ ...
2
votes
0
answers
114
views
Is there an explicit version of Morse Lemma used in stationary phase method?
In the proof of the stationary phase method (at least the one I have seen) Morse lemma shows up, which states: Let $g:\mathbb R^n\to \mathbb R$ be a function of class $C^\infty$ for which $0$ is a ...
- 3,625
1
vote
1
answer
250
views
Question about the stationary phase method and the smooth function used
A statement of the stationary phase method I know is the following.
Suppose $\phi(x_0) = \phi'(x_0) = 0$ and $\phi''(x_0) \not = 0$. If $\psi$ is a smooth function supported in a sufficiently small ...
- 3,625