Complex Analysis: Integral of (x^n)/(x+1) using Contour Integration

Today, we evaluate the improper integral from 0 to infinity of (x^n)/(x+1) where n is a negative real number between -1 and 0. We will be using this result in a future video!

[nyu]

it’s been 30 years since I have evaluated an integral like this. thanks for the trip down memory lane !

[FunctionalIntegral]

Super nice and interesting. Please do more such complex integrals along different and crazy paths.
I like very much THIS camera position. Please don't change it in future tutorials.

[Frutyy]

Your explanation on the relationship between the integrals psi_1 and psi_2 helped a lot . Perhaps I’ll have a better understanding of how integrals similar to those two behave now 😂

[dewman]

Thanks man. Another great video. This is a very similar example to help me on my homework.

[syedzaidi]

The integrand does not have a pole at z=0. That is a branch point. We still need to examine the integral (little gamma), which vanishes.

[lordstevenson]

Wow. Just wow.
This has motivated me to take a Complex Analysis module during my degree, wish me luck 😂

[quentinrenon]

You can also use the beta function on the original integral, and then Euler's reflexion formula to instantly find the final answer

[Ravi-qzof]

You have explained this topic masterfully. Cheers!

[rahulnadig]

That’s a beautifully made video. Keep it up

[vesperflix]

I am searching for this for a long time

[eric]

You really make great Videos, i really find them interesting. Keep up the great work! :D

[joemckeon]

Could you please do an example with the branch cut along the positive imaginary axis? Great vid btw

[josephhajj]

This is a hard path to think that it will fit this equation

[hydropage]

I don’t understand how the e^2pi*i*n doesn’t become (e^2*pi*i)^n, becoming 1^n, which would make phi 2 equal to negative of the original integral. At 38:16, you say that e^2*pi*i is just 1, so you ignore that, but why didn’t that apply to the numerator of the integral? You should’ve done that there too, right? Why did that n get carried all the way to the end

[domc]

Just did this in 15 mins with feynmann trick define, I(t) = int( e^(-t(x+1)) x^n/x+1.

You end up with gamma(n) gamma(1-n) which simplifies by eulers reflection formula

[michaelredenti]

Why just not draw the horizontal segment \psi_1 up until the intersection with the y-axis? Then you do not even have to consider delta. Am I missing something?

[runnalex]

Hello, I got a question, I understood all the past videos on Complex Integrals but there wasn't branch cuts, why is there one branch cut here ? Is it because it was always pure imaginary poles and we didn't have to worry about the real axis ? And why could you just take the limit as epsilon approches zero in the first place when it was above the real axis while you had to use the ln when you were below the real axis ?

[xulq]

idk why but the factor of -e^(2*pi*i*n) at 41:43 kinda already intuitively came into my mind before even watching you calculate it by hand. i was just thinking if psi_2 goes to the other direction we just multiply the integral by exactly that factor to get capital I. anyone else?

[shandyverdyo]

Hi.
At 30:11 you wrote the parametrization as follows:
z=t+iε, t€[R, ε] but the interval means R≤t≤ε and it's uncomfortable to look at, since R>ε.
Why it's not:
z=-t+iε, t€[-R, ε] ?

[bebarshossny]

bit confused, for easier integrals of this sort say from -infinity to infinity we draw a line on the whole x axis just fine. why in this special case did we start considering the branch cut