Complex Analysis: Integral of 1/(x^n+1) feat. pizza contour

Today, we revisit an old classic on the channel, the integral from 0 to infinity of 1/(x^n+1) where n is any real number greater than 1.

[qncubed]

Note: Typo at 3:55 should be an element symbol instead of equality ... silly me

[wuyqrbt]

Without any doubt:
You're *The King Of Complex analysis* on YouTube.
Please continue this playlist.
Thank you 💖

[davidblauyoutube]

When I first did this integral and got the right answer, I knew finally that I really understood complex analysis.

[mohamedkhoulali]

this channel is so f underrated ! .. the best on complex analysis thank you

[wuyqrbt]

*Happy first contour integral with chalk board*
Yeah, I watched a whiteboard version of it before, but with some difficulty. But this one is great, in all aspects.
And ...
Please when you are busy, at least make short videos.
Thank you so much dear *QN³* ❤️

[itisajem]

Interesting the result looks like the reflection formula for the gamma function but with 1/n

[Circuito]

A question: residue method can only be used to calculate definite or improper integrals but not for indefinite in order to obtain only the primitive?

[rayandy]

video! However, can n be non-integer?

[laurimynttinen]

Can you make a video explaining contour integrals?

[achenejegodwin]

Thank you for that wonderful piece of delivery, pls, can you help when n=5, I.e f(x) = 1/ x^5 + 1

[calebkan]

all u need is the beta function then put into gamma form and use euler's reflection formula

[bleaks]

An interesting, alternative form for the final answer:

I = (1/n) * Γ(1/n) * Γ(1-1/n)
I = Γ(1+1/n) * Γ(1-1/n)
I = B(1+1/n, 1-1/n)

[ryanblais]

Thanks for this great video and explanation. Just a question, at 7:16, should there be two or three poles in the lower right quadrant (positive Real, negative Imaginary)?

[TheHellBoy]

A much simpler aproach, about how i solved it. Substitute x=t^1/n

This makes dx=t^((1/n)-1)dt

The given function resolves to the form of beta function. Which later simplifies into eulers reflection formula

[wuyqrbt]

We are waiting
🧐 It's me, looking at screen, for your notification 🧐

[Nolord_]

That's pretty nice. Would it be possible to generalize this result for R=1?

[lambda]

There’s actually a better method divide the denominator and numerator with x^n and then apply partial fraction and then resolve the contour

[holyshit]

I would probably calculate it with Beta function then change it to Gamma function, finally i would finish it with reflection formula for Gamma

[bonelesspizza]

But why are you allowed to choose a contour that's only around a single pole? Why not choose a contour that encloses 2 poles? How diff would the answer be?

[ayman]

What if we replaced n by 5, how will the integfation be, and what will the answer be??

Do i just replace n by 5 in all the steps of the solution and in the final answer or what??