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Mar
20
comment Expansion of a function analytic at infinity
Perhaps try the mapping $z \mapsto \frac{1}{z}$?
Mar
20
revised Proving $\frac{\cos(t \arctan(\sqrt{x}))}{(1+x)^{t/2}}= \sum_{k \ge 0}\frac{\Gamma(t+2k)\Gamma(k+1)}{\Gamma(t)\Gamma(2k+1)}\frac{(-x)^k}{k!}$
added 155 characters in body
Mar
19
accepted Ramanujan's 'well known' integral, $\int_\frac{-\pi}{2}^\frac{\pi}{2} (\cos x)^m e^{in x}dx$.
Mar
19
asked Proving $\frac{\cos(t \arctan(\sqrt{x}))}{(1+x)^{t/2}}= \sum_{k \ge 0}\frac{\Gamma(t+2k)\Gamma(k+1)}{\Gamma(t)\Gamma(2k+1)}\frac{(-x)^k}{k!}$
Mar
19
comment Ramanujan's 'well known' integral, $\int_\frac{-\pi}{2}^\frac{\pi}{2} (\cos x)^m e^{in x}dx$.
If I was much cleverer, I would try to express $\frac{\pi}{2^m}$ as a product of two integrals, multiply both sides by $\Gamma \left(1+ \frac{m+n}{2}\right)\Gamma \left(1+ \frac{m-n}{2}\right)$ and rewrite both triple integrals so that their equality is obvious. I've not been able to come up with a solution. I've also tried the beta function, to no avail.
Mar
19
asked Ramanujan's 'well known' integral, $\int_\frac{-\pi}{2}^\frac{\pi}{2} (\cos x)^m e^{in x}dx$.
Mar
18
comment Infinite series
@OJB See the Wikipedia article.
Mar
17
comment Isomorphism between this subgroup of complex numbers and all finitely generated abelian groups ?
@IanColey wasn't aware of this, quite possibly.
Mar
17
asked Isomorphism between this subgroup of complex numbers and all finitely generated abelian groups ?
Mar
1
awarded  Notable Question
Feb
26
comment Differentiation wrt parameter $\int_0^\infty \sin^2(x)\cdot(x^2(x^2+1))^{-1}dx$
Are we able to use $\int_0^\infty \frac{\cos(kx)}{x^2+1}dx= \frac{\pi}{2e^k}$ as a given?
Feb
26
revised Bessel functions: proof that $J_0(z)=\frac{1}{\pi}\int_0^\pi e^{i z \cos(\theta)}d \theta$.
typo fixed
Feb
26
suggested suggested edit on 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 $J_0(z)=\frac{1}{\pi}\int_0^\pi e^{i z \cos(\theta)}d \theta$.
Thank you. I know little about Bessel functions and this helped greatly!
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
8
comment Evaluate $\int_0^\infty e^{-x^4}x^2dx$
$u=x^4$, then use the Gamma function.