Complex Analysis: Integral of xsin(x)/(x^2+1) using Contour Integration

Today, we use contour integration and the residue theorem to evaluate an improper integral.

Laplace Transform Method:

[Kzey]

Hi, I don't understand why do you replace sin(z) by e^iz in the begining ? Because sin(z)=(e^iz-e^-iz)/2i ??? I know it's correct but I don't understand :)

[KostasGiotis]

Hello sir, and thank you for this video.

I wanted to try solving this by writing sin(z) as (e^iz - e^-iz)/2i and then calculating the residue of this new f(z) at z=i which turns out to be Res(f(z), z=i) = (i/4)*(e - 1/e).
Then I = 2πi*Res(f) = 2πi*(i/4)*(e - 1/e) = -(π/2)*(e - 1/e) which is not the same as your answer!

Could you tell me where the mistake is?

[ozzyfromspace]

Complex analysis is such an amazing thing! I’m watching all your videos to get exposure to the main ideas, so I can link the theory to practice. Thank you for doing this, I’m feeling better about this stuff now, less intimidated by it

[HuntBobo]

Great videos. Didn’t you drop the I you brought out front around 10 minutes?

[lorenz]

How do we explain after all why can we guarantee the existance of the integral (and so its equality to Principal Value and limR->oo of the integral in [-R, R])? Usually, we proove that f is bounded by a riemann integrable function (which is more obvious to show) and so we have the existance of f integral ... but I can't seem to find a riemann integrable bound for this one, any ideas????

[HarshKumar-kxqy]

sir, is this integral twice that of integral done from 0 to infinity?

[gaaraofddarkness]

Such a complex thing explained simply. Thanks mate

[akmarzhanabilassan]

Very good explanation.This helped a lot. Thank you!

[geraldnavida]

Hello, thanks for this video. I just have one question. If we evaluate the limit of an integral, we can only interchange the order of limit and integration if the integrand is a monotonically increasing function and measurable according to monotone convergence theorem, right? The integrand e^-Rsint is measurable but it is a decreasing function as R becomes larger. Do you think you can apply monotone convergence theorem here? Another theorem I think that can be invoked is Lebesgue Dominated Convergence Theorem if MCT won't work.

[jjzhu]

why do you need to do e^iz and take imaginary instead of doing sin(z) directly

[wuyqrbt]

Thanks a lot dear *QN³*
Now I'm so interesting about the Laplace one.

[vahiback]

Sir what is answer if the denominator is(x^2-r^2).
Can you help me sir. If r is real

[heattransfer]

so we replaced sin(x) by e^iz. Can I also replace cos(x) by e^iz too ?

[LatifaEssa]

Pourquoi on ne peut pas trouver des vidéos comme ça en Francais 😢
Merci beaucoup tu es incroyable

[dewman]

How do you generally choose your contour in a given integral? If I choose a different contour, will I get another answer? What's the thought process? It's the hardest part