Contour Integration #6 - Integrating log(x)/(x^2 + 1) - LearnMathsFree

NOTE: I made an error when taking the limit of arctan(x) as x approaches infinity. It should be pi/2, not pi/4. The imaginary parts of both sides would then cancel. (But it doesn't matter, because the integral we were evaluating was purely real.)

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In this video, we look at a more involved example of a contour integral. We explain how to take branch cuts using principal and non-principal definitions of the complex logarithm Log(z). We evaluate the integral using the method of contour integration, by using an indented semi-circular contour. We then use the residue theorem, together with the estimation lemma, to derive the required result. More to follow.

[PunmasterSTP]

log(x)/(x^2+1)? More like "Lots of good lectures, and watching them in succession can be fun!" 👍

[federicovolpe]

you can also obtain the result by real methods by substituting (1/u)=x. You obtain that I=-I and therefore the whole thing is 0. Anyway, nice video ;)

[rickybobby]

at time 31.06 its pi/2 not pi/4 . great videos

[ErcanErgul]

Great video to remember the old times. :)

[NoFancyTitlesNeeded]

Great video don't stop making them!

[ozzyfromspace]

Your explanation is perfect for students of complex analysis ☺️🎊🙌🏽 Thanks mate, this gave me an idea for how to solve a similar problem for some math research I’m doing 🙏🏽💯

[santiagoerroalvarez]

Nice vid. Gotta point out that |ln|z|| in the gamma sub r case is equal to |ln(r)| but because 0<r<1, ln(r)<0, and |ln(r)| = -ln(r) ≠ ln(r). Thankfully, that didn't affect the final result :D

[AM_JOSE]

Great video!!..suppose i have: log x/ (x² +1)² then how should I solve it??

[mrJianth]

I really enjoyed this video, very good explanation. My lecturer did the same problem but via a keyhole integral using log(z)^2/(1+(-z)^2), and that method was quite confusing. What if i had the same integral but instead of x^2 we had x^3, i.e. log(x)/(1+x^3)? What would be the best method to set up the problem?

[lautaro]

excelente jobs.. Thank you a lot for your help! :)

[rickybobby]

awesome videos thank you but can you post videos on more contour integrations like dog-bone and key-hole contours?

[abdullahtan]

what happens when we have log(x-3) instead of logx ?

[zevkap]

arctan(-infinity) = -\pi/2. Otherwise, your explanation is perfect

[utrigos]

Why do you add Pi to be the largest possible argument when you rewrote the argument bounds to be from -pi/2 to 3pi/2? shouldn't it be 3pi/2?

[Anders]

Let's say I have ln(x)/(x^2-1) from 0 to infinity. I already know how to evaluate it, but can I use this method also? If I solve x^2-1=0 I obtain real solutions. Not complex.