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May
4
revised Asymptotic related to the infinite product of sine
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May
3
revised Asymptotic related to the infinite product of sine
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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.
May
2
comment Snags when discovering the asymptotic behavior of an integral
Huh, since these days I read part of de Bruijn's, I picked some stuff out from my mathematical analysis textbook and tried to determine the asymptotic behavior. Unfortunately, I found the road is extremely distorted. This one is just one of these problems. I've posted a new question related to the proof of the infinite product of sine. I will be pleased if there's some asymptotic analyst (professor) around me.
May
2
comment Snags when discovering the asymptotic behavior of an integral
I doubt that it could be expanded. I guess the term after $Cn^{-2}$ is $O(n^{-3}\log n)$, not $O(n^{-3})$.
May
2
comment Snags when discovering the asymptotic behavior of an integral
Typo, the remainder is $\displaystyle\frac1{n^2\pi^2}\int_0^{\pi/2}\frac{x\sin xdx}{(\sin x+1/n\pi)(\sin x+(1+x\sin x)/n\pi)}$.
May
2
comment Snags when discovering the asymptotic behavior of an integral
The remainder is $\displaystyle\frac1{n^2\pi^2}\int_0^{\pi/2}\frac{x\sin xdx}{\sin x+1/n\pi}{\sin x+(1+x\sin x)/n\pi}\asymp\frac1{n^2\pi^2}\int_0^{\pi/2}\frac{xdx}{\sin x}$, where the absolute error is $\displaystyle-\frac1{n^3\pi^3}\int_0^{\pi/2}\frac{(2\sin x+x\sin^2x+(1+x\sin x)/n\pi)xdx}{(\sin x+1/n\pi)(\sin x+(1+x\sin x)/n\pi)\sin x}$, which is not easy to determine the major part (it seems that Lebesgue's dominated convergence theorem doesn't work well).
May
2
comment Snags when discovering the asymptotic behavior of an integral
Well, I'll compute it soon. Incidentally, does some method called saddle point method work for this integral? I have no idea about complex analysis and Cauchy integral theorem. It is said that such a method is very powerful in de Bruijn's Asymptotic methods in analysis.
May
1
asked Asymptotic related to the infinite product of sine
Apr
30
comment Snags when discovering the asymptotic behavior of an integral
I wonder whether we could look into $O(1/n)$. The $\sin$ function is very nasty. The major distribution of the integrand is near $0$, but we couldn't decompose the interval $[0,\pi/2]$ easily to obtain the result.
Apr
30
comment How to interpret little-o notation in an exponent.
A systematical way to interpret these phenomena is to consider $o$ and $O$ sets of functions. For example, $o(g(n))=\{f(n):\forall\epsilon>0,\exists N,\forall n\ge N,\lvert f(n)\rvert<\epsilon\lvert g(n)\rvert\}$, and $O(g(n))=\{f(n):\exists M,\forall n,\lvert f(n)\rvert\le M\lvert g(n)\rvert\}$, and $f(n)=o(g(n))$ is conventional abbreviation for $f(n)\in o(g(n))$.
Apr
29
comment Asymptotic for the integral involving exponential
Since $\exp(x^n)=\sum_0^\infty x^{nk}/k!$ converges absolutely on $[0,1]$, the interchange of summation and integration is justified. The interchange of summation is valid for the absolute convergence of the double sum. Am I right?
Apr
29
accepted Asymptotic for the integral involving exponential
Apr
29
comment Asymptotic for the integral involving exponential
Very good, thanks!
Apr
29
revised Asymptotic for the integral involving exponential
added 70 characters in body
Apr
29
comment Does $ \int_0^{\infty}\frac{\sin x}{x}dx $ have an improper Riemann integral or a Lebesgue integral?
One could calculate $\int_0^\infty\sin xdx/x$ elementarily. It's equivalent to $\lim_{n\to\infty}\int_0^{\pi/2}\sin2nxdx/x$. We're pleased to see that $g(x)=1/x-1/\sin x=O(1/x)$ and so $\lim_{n\to\infty}\int_0^{\pi/2}g(x)\sin2nxdx=0$ (Generally, Riemann-Lebesgue lemma; however, since $g$ is of $C^\infty$, we could integrate it by part and obtain the result). Since $\sin2nx/\sin x=\sum_{k=1}^n(\sin2kx-\sin2(k-1)x)$, we could easily determine $\int_0^{\pi/2}\sin2nxdx/\sin x$.
Apr
29
comment Compactness proof of $\mathbb{R}^2$
It could be rather general if we come into the situation of topology spaces. The proof is similar in such a way: if $Y$ is compact and a open set of $X\times Y$ contains $x_0\times Y$, then there's a neighborhood $U$ of $x_0$ in $X$ such that $V$ covers $U\times Y$. For detailed information, see wiki or Munkres' topology.
Apr
29
asked Asymptotic for the integral involving exponential
Apr
27
asked Snags when discovering the asymptotic behavior of an integral
Apr
12
comment The equivalence between Cauchy integral and Riemann integral for bounded functions
Let $\omega(x_0)=\lim_{u\to0}(\sup f(x)-\inf f(x))$ be the oscillation around $x_0$, we have the discontinuities $D=\cup_n D_n$ where $D_n=\{x\colon\omega(x)\ge 1/n\}$ are compact subsets, therefore $m(D)=0$ if and only if $m(D_n)=0$ for all $n$, and note that $m(D_n)$ is just the content, not only the measure. The characterization about $D_n$, of Riemann-integrability for bounded functions, which could be proved elementarily, is more and more prevalent on the calculus books.