2,022 reputation
1524
bio website
location
age
visits member for 1 year, 7 months
seen 12 hours ago

Jan
31
comment Evaluating $\int_{0}^{\infty} \frac{x^{3}- \sin^{3}(x)}{x^{5}} \ dx $ using contour integration
(I'm assuming you're using a semicircular contour). A somewhat related question: how would one justify that $\lim_{R \to \infty}\left|\int_0^\pi \frac{e^{i3Re^{it}}-3e^{iRe^{it}}}{(Re^{it}-i \epsilon)^5}\right|=0$?
Jan
29
asked Ramanujan's partial fraction decomposition of $\frac{1}{(x^2+a^2)\cdots(x^2+(a+n)^2)}$.
Jan
20
answered Convergence of the integral $\int\limits_{1}^{\infty} \left( \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x+3}} \right) \, dx$
Jan
17
comment Calculating the number of possible paths through some squares
It may be worth noting for the OP that a similar method works if $x,y$ are in $n$-dimensional space.
Jan
15
comment Integral Of $\int \frac{\cos(3x)}{(x^2+1)^2}dx$
Use a semicircular contour in the upper half of the complex plane. Show that the curvy part of the integral goes to $0$ at the radius $\to \infty$. The semicircular contour integral is equal to $2 \pi i Res_{z= +i} \frac{\cos(3z)}{(z^2+1)^2}$.
Jan
13
comment South Africa National Olympiad 2000 (Tile 4xn rectangle using 2x1 tiles)
Surely $A_2=4$?
Jan
4
comment Why is the equality true: $\int_{u=0}^1 u^{\alpha_1-1} (1-u)^{\alpha_2-1} \, du =\frac{\Gamma(\alpha_1)\Gamma(\alpha_2)} {\Gamma(\alpha_1+\alpha_2)}$
en.wikipedia.org/wiki/…
Dec
29
revised Finding the number of distinct $m$-tuples of distinct numbers whose product is $n$.
added 4 characters in body
Dec
28
comment Finding the number of distinct $m$-tuples of distinct numbers whose product is $n$.
@benh Thanks for pointing that out.
Dec
28
revised Finding the number of distinct $m$-tuples of distinct numbers whose product is $n$.
edited title
Dec
28
revised Finding the number of distinct $m$-tuples of distinct numbers whose product is $n$.
edited title
Dec
28
asked Finding the number of distinct $m$-tuples of distinct numbers whose product is $n$.
Dec
27
comment Closed form for $\sum_{n=-\infty}^\infty \frac{1}{(z+n)^2+a^2}$
@RonGordon are there any restrictions on $f(n)$ so that the first result holds? It seems nonsensical if $f(n)$ is entire.
Dec
21
comment Why does $\lim_{x \rightarrow 0} B(x,y)$ exist and how is it calculated?
Thank you. In this case, is it possible to easily calculate the integral directly from the beta limit, or is the above method much easier?
Dec
21
comment Understanding $n \left(\frac{2n \choose n}{4^n}\right)^2$ for large $n$
en.wikipedia.org/wiki/Central_binomial_coefficient#Properties See property $2$.
Dec
21
accepted Why does $\lim_{x \rightarrow 0} B(x,y)$ exist and how is it calculated?
Dec
20
comment Why does $\lim_{x \rightarrow 0} B(x,y)$ exist and how is it calculated?
Thanks very much for answering thoroughly. So is the crux of the existence of the limit that although the limit of a function may exist, the limit of its derivative may exist?
Dec
16
comment Why does $\lim_{x \rightarrow 0} B(x,y)$ exist and how is it calculated?
@gammatester Thank you, although my question boils down to finding a nicer form for that limit and (even if the prior isn't possible) to prove that the limit equals $\frac{-1}{2}\zeta(4).$
Dec
16
comment Why does $\lim_{x \rightarrow 0} B(x,y)$ exist and how is it calculated?
@gammatester wolframalpha.com/input/…. It seems that the integral is equal to the limit, although the factor of $2$ before $2I$ was incorrect.
Dec
16
revised Why does $\lim_{x \rightarrow 0} B(x,y)$ exist and how is it calculated?
added 194 characters in body