$\int^{\infty}_{0}\frac1{x^n+1}dx$
My current method is contour integration, with the contour being a sector with a radius R (R goes to infinity); sector has its corner at 0. The contour contains only one nth root of unity within it
#Integration help
rich shell
versed hillBOT
No Lifer#GTFOanemia
next elmBOT
$\int^{\infty}_{0}\frac1{x^n+1}dx$
My current method is contour integration, wi...
- Wait patiently for a helper to come along.
- Once someone helps you, say thank you and close the thread with:
+close
- Feel free to nominate the person for helper of the week in #helper-nominations
- Do not ping the mods, unless someone is breaking the rules.
- If you're happy with the help you got here, and the server overall, you can contribute financially as well:
rich shell
my work so far:
fathom python
Hm. Well, I don't know complex analysis, but I have derived the following a while ago.
rich shell
$\oint_{\alpha}\frac1{z^n+1}dz = 2i\pi Res(f,e^{\frac{i\pi}{n}})$
(Take alpha as the contour which I'm integrating along and f as the weird function I'm integrating)
$\oint_{\alpha}\frac1{z^n+1}dz = 2i\pi\lim_{z\to e^{\frac{i\pi}{n}}}\frac{z-e^{\frac{i\pi}{n}}}{z^n-1}$
(I got stuck evaluating the limit)
$\oint_{\alpha} = \int^{\infty}{0}\frac1{z^n+1}dz + \int{\Gamma}\frac1{z^n+1}dz + \int_{l}\frac1{z^n+1}dz$
(Take capital gamma to be the curvy part of the sector and l to be the tilted line part of the sector)
The integral along gamma then goes to zero as R goes to infinity (Benefits of abusing Jordan's Lemma)
$\oint_{\alpha} = \int^{\infty}{0}\frac1{z^n+1}dz + \int{l}\frac1{z^n+1}dz$
fathom python
You can just type it.
I'm sure people will understand.
If you need symbols, I can provide you what you need.
rich shell
If you need symbols, I can provide you what you need.
uh there's a lot of symbols but dw i can fix the latex
Thx anyways though
swift oliveBOT
@rich shell has given 1 rep to @fathom python
fathom python
Ah, ok.
versed hillBOT
No Lifer#GTFOanemia
rich shell
that looks horrendous-
I sincerely apologize to anyone reading this
fathom python
Why? Looks nice to me.
Or did you mean the content? In that case, can't say anything about it.
rich shell
Why? Looks nice to me.
Or did you mean the content? In that case, can't say anyt...
it looks nice to you huh... the content is on the limit of understanding tbh but anyone experienced should know what I'm trying to convey
Also I saw your solution
Factoring the integrand...
I'll need some time to understand it tbh
fathom python
it looks nice to you huh... the content is on the limit of understanding tbh but...
My solution is basically just partial fractions.