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May
11
comment how to prove $f$ is an arithmetic function with this property $\sum_{d\mid n} f(d)=n^2$
I think it's a direct result of Möbius inversion formula
May
10
comment Absolute values in $\int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}$
Another approach: Let $u=x+2$, we need to integrate $du/u\sqrt{u^2-1}$. Set $v=\sqrt{u^2-1}$, we have $du/u=vdv/(v^2+1)$ and $dx/(x+2)\sqrt{(x+1)(x+3)}=dv/(v^2+1)$, therefore the answer is $\arctan v$ where $v=\sqrt{(x+1)(x+3)}$.
May
10
comment Conditions that allow Integration by Substitution
By minimum, you mean the necessary and sufficient condition of $\phi$ such that for each $f$ is (Riemann/Lebesgue)-integrable on $[a,b]$, we have $\int_a^b f(\phi(x))\phi^\prime(x)dx=\int_{\phi(a)}^{\phi(b)}f(x)dx$. It's quite hard. I know that it's true when $\phi$ is monotone.
May
10
comment Conditions that allow Integration by Substitution
$\phi$ needn't to be continuously differentiable. For example, if $\phi$ is monotone and differentiable, where $\phi^\prime$ is integrable, the theorem is also correct. @ήλιος
May
10
comment Asymptotic related to the infinite product of sine
@AntonioVargas I think I've got the key to attack the problem. Substract $S_n$ with $\sum_{k=1}^n\ln(1-x^2/k^2\pi^2)$, and estimate the summation through Euler-Maclaurin formula. You can plot a graph for the summation, and find that the distribution is very flat, which case is suitable for Euler-Maclaurin! Detailed calculation is not performed since the calculation is tedious.
May
5
comment How prove this linear algebra $AB=BA$?
It is also related, Motzkin & Taussky's original proof.
May
5
comment How prove this linear algebra $AB=BA$?
I hope if there's some pure elementary approach.
May
5
comment How prove this linear algebra $AB=BA$?
The second link seems unavailable.
May
5
revised Tricky elementary integral
added 147 characters in body
May
5
answered Tricky elementary integral
May
5
comment Snags when discovering the asymptotic behavior of an integral
Although it's not a complete answer, it's well-informed and informative.
May
5
accepted Snags when discovering the asymptotic behavior of an integral
May
4
awarded  Announcer
May
4
comment Laplace integration after the first term
Bibliography: de Bruijn's Asymptotic methods in analysis
May
4
revised Prove an integral limit
edited body
May
4
comment Laplace integration after the first term
The substitution $t=h(s)$ does work; however, the integral become an improper integral. Note that the improper integral $\Gamma(1+\alpha)=\int_0^\infty e^{-q}q^\alpha dq$ converges even when $-1<\alpha<0$.
May
4
answered Prove an integral limit
May
4
revised Asymptotic related to the infinite product of sine
added 273 characters in body
May
3
revised Asymptotic related to the infinite product of sine
added 356 characters in body
May
3
comment Asymptotic related to the infinite product of sine
@AntonioVargas $x$ is a constant, as I've said. What I really need, is methods, discipline to deal with such a summand. Comparing is just yielding by-products.