Complex Analysis: One EXOTIC Integral

Today, we attempt to integrate a function with infinitely branch cuts. Integral of x*log(1+e^x)/(1+x^2)^2 from -infinity to infinity

[lordstevenson]

Would love to see more examples using residue theorem on higher order poles ❤️

[danielesantospirito]

Exotic indeed! By the way, I were you, I wouldn't be able to sleep tonight without checking first that the infinite sum of the small contour pieces on the left side is zero... (they weren't even given a name, that's sad)

[dihinamarasinghe]

Great integral and even better I was wondering if you could upload a video on arguments of functions like 1+e^z or

[Decrupt]

Haven’t watched it yet, but this is going to be a banger. Love these exotic variants!

[lordstevenson]

You know the videos a banger when the contour looks like an alien robot.

[SiphonSoulsX]

Very cool integral, I had solved it by myself if not for that cheeky tangent expansion, I think that a work-around in terms of complex exponentials could be found with a bit of time

[holyshit]

Integration by parts + odd and even decomposition of function
and this integral is accessible also for beginners in calculus who dont know any advanced techniques for calculating integrals

[TheRandomFool]

Definitely interesting to see how this integral was done as-is. Integration by parts before contour integration does make it a fair bit easier, though. The same series shows up in this method too, but is attached to the imaginary part which gives the result of the series. The real part is the result of the desired integral.

[trannam]

Why in 7:16 since 1+e^x*cos(y)>0, you can draw such parabolic graph which defined on positive value. And how could you define the direction of the area on that Re(_) plane?

[milkymilsy]

boi i haven't watched the vid yet but ik this gonna be awesome

[푸른하늘-bz]

Great!, thank you for your lecture,
would you explain series of tan(z)(31:17)? I don't know result of this series.

[NoNo-oeft]

Do you just stumble across these mad intergrals?

[dihinamarasinghe]

Qucubed3 can you do a video on cauchy principle value integrals and can you explain this....now you know that we can evaluvate small epsilon radius circular contours by using residues right ?...but in principle value problems the actual epsilon radius contour around the pole referes to the principle value while the residue referes to the actual integral (not the principle value) why does this happen can anyone explain [ try evaluvating 1/z^2 from -infinity to infinity using complex analysis ]you'll see what I mean...

[푸른하늘-bz]

thank you, I ask you one more series expansion of tan(z) 31:17 I want to learn principle of this series of tan(z).

[sophiaxiao]

Hi! Would you consider doign videos of real analysis? I've never seen someone explain proofs so clearly and thoroughly, except maybe some genius guy raised by a professor.

[jieyuenlee]

11：09why not using box contour with height 2pi? then it only enclosee i and ipi

[him]

Not to be a party pooper, but I'd like to present a much simpler method. Let's call this integral J. Taking x->-x finds dx. So J = (1/2)(J + J) = dx and because (1+e^x)/(1+e^{-x}) = e^x and log(e^x)=x, this means dx which is an elementary integral that can be evaluated by direct antiderivative evaluation or by the residue theorem (but with a much simpler contour)

[mattmiller]

Integral of exp(-x^2/2)*cos(a*x)/(x+b) from -infinity to infinity

Thoughts? 😅

[daddy_myers]

Babe wake up, qncubed3 just dropped an absolute banger.