14,948 reputation
12774
bio website
location
age
visits member for 3 years, 2 months
seen yesterday

1d
awarded  Good Question
2d
revised Prove that $\int_0^\infty \frac{e^{\cos(ax)}\cos\left(\sin (ax)+bx\right)}{c^2+x^2}dx =\frac{\pi}{2c}\exp\left(e^{-ac}-bc\right)$
I added a missing detail and added a link
2d
comment Prove that $\int_0^\infty \frac{e^{\cos(ax)}\cos\left(\sin (ax)+bx\right)}{c^2+x^2}dx =\frac{\pi}{2c}\exp\left(e^{-ac}-bc\right)$
@Venus The Taylor expansions of $\frac{1}{1 - z}$ and $\frac{1}{1+z}$ at the origin don't converge for values on the unit circle. They only converge if $|z| <1$. And $\frac{e^{z}}{z}$ is not complex differentiable at the origin. So it doesn't have a Taylor expansion there.
2d
comment Prove that $\int_0^\infty \frac{e^{\cos(ax)}\cos\left(\sin (ax)+bx\right)}{c^2+x^2}dx =\frac{\pi}{2c}\exp\left(e^{-ac}-bc\right)$
@Venus That the radius of convergence of the Taylor series is exactly $1$ or greater. Otherwise I wouldn't be able to replace the complex variable $z$ by $e^{iax}, x \in \mathbb{R}$. Functions that satisfy this condition include $e^{z}, \cos(z), \sin(z)$, and $z^{n}$ where $n$ is a positive integer. An example of a function that doesn't is $\frac{1}{\sqrt{1-4z}}$.
Nov
20
revised The integral $\int_{|z|=2}\log\frac{z+1}{z-1}dz$
added 13 characters in body
Nov
20
revised The integral $\int_{|z|=2}\log\frac{z+1}{z-1}dz$
added 284 characters in body
Nov
20
comment The integral $\int_{|z|=2}\log\frac{z+1}{z-1}dz$
@PedroTamaroff It holds because as Daniel Fischer pointed out to me, $1- \frac{1}{z}$ and $1 + \frac{1}{z}$ lie in the right half-plane for $|z| >1$. I would just let $z=x+iy$ to see that's the case. Thus $ - \pi < \text{Arg} \left(1 + \frac{1}{z} \right) - \text{Arg} \left(1- \frac{1}{z} \right) \le \pi $.
Nov
20
revised The integral $\int_{|z|=2}\log\frac{z+1}{z-1}dz$
added 15 characters in body
Nov
20
comment The integral $\int_{|z|=2}\log\frac{z+1}{z-1}dz$
@PedroTamaroff $\log \left( \frac{1 + \frac{1}{z}}{1- \frac{1}{z}} \right) = \log \left( 1+ \frac{1}{z} \right) - \log \left(1 - \frac{1}{z}\right) $ and then expand both functions in Taylor series
Nov
20
answered The integral $\int_{|z|=2}\log\frac{z+1}{z-1}dz$
Nov
18
revised Calculate the residue of $\cot\pi z$ at poles $z=n$
corrected a misstatement
Nov
18
awarded  Necromancer
Nov
18
revised Solving an integral equation using the Fourier transform
edited title
Nov
18
revised Solving an integral equation using the Fourier transform
improved really bad formatting
Nov
17
revised Compute $\int_0^\infty \frac{\ln x}{(1+x)^3}\,\mathrm{d}x$
improved formatting
Nov
17
revised Integrating around pie-slice domain
improved formatting
Nov
17
revised Keyhole Domain Residue problem with logarithm
improved formatting
Nov
17
revised Inequality with Gamma function: how to prove it?
An equals sign was missing and I improved formatting
Nov
15
awarded  Nice Answer
Nov
15
revised What is the value of $\int_{0}^{\infty} \frac{x \ln (x^{2}+1)}{\sinh (\pi x)} \ dx $?
made the title more descriptive