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Feb
23
accepted Bessel functions: proof that $J_0(z)=\frac{1}{\pi}\int_0^\pi e^{i z \cos(\theta)}d \theta$.
Feb
23
comment Bessel functions: proof that $J_0(z)=\frac{1}{\pi}\int_0^\pi e^{i z \cos(\theta)}d \theta$.
@CameronWilliams I'm happy for a prospective answer/comment to use any widely-used definition. To obtain $I(z)$ I used the differential equation definition.
Feb
23
revised Bessel functions: proof that $J_0(z)=\frac{1}{\pi}\int_0^\pi e^{i z \cos(\theta)}d \theta$.
edited title
Feb
23
asked Bessel functions: proof that $J_0(z)=\frac{1}{\pi}\int_0^\pi e^{i z \cos(\theta)}d \theta$.
Feb
22
asked A nice form for this Beta-like integral $\int_0^\frac{\pi}{2} \sin^\alpha(n t) \cos^\beta(t)dt$?
Feb
7
comment Area of a triangle using vectors
$\mathbf{a}\cdot \mathbf{b}=|\mathbf{a}||\mathbf{b}| \cos \theta$.
Feb
1
accepted Ramanujan's partial fraction decomposition of $\frac{1}{(x^2+a^2)\cdots(x^2+(a+n)^2)}$.
Feb
1
revised Ramanujan's partial fraction decomposition of $\frac{1}{(x^2+a^2)\cdots(x^2+(a+n)^2)}$.
edited body
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 $n$.