Integral with oscillation

Integral of (1-cos(x))/x^2 from 0 to infinity

In this neat video, I calculate the integral from 0 to infinity of 1-cos(x)/x^2 using complex analysis. For this I choose a very clever contour that jumps over the singularity at 0. This problem is taken from the book Complex analysis by Stein and Shakarchi

[chatop]

You and BlackPen RedPen are my favorite Math Youtubers :)

[pcmanpacmn]

Watching it with my morning coffee. Thanks.

[TheOskay]

I had a course on complex analysis back in January at Uni, the ideas are really elegant and beautiful! Thanks for making this video!

[ozzyfromspace]

Thank you for thIs video! Watching you do a complex analysis problem makes me feel more comfortable giving this a try as someone who is learning on their own 🙏🏽🥳

[FT]

I don't know much complex analysis, but this was still clear and well explained. Lots of steps seem hard to come up with though!

[mariadelcarmenjimenez]

I like how you explain. Hello from Rep.Dom.

[AbduraufQodiriy]

So cool) Complex analysis is just amazing

[edmundwoolliams]

Physicist method: 1-cos^2(x) is approximately x^2/2. So the integrand = 1/2 and so the integral diverges

[tahirimathscienceonlinetea]

It’s a beautiful integral keep up bro

[josephhajj]

Dr peyam you can solve it using Jordan lemmas for the circular contour

[adamlimani]

Actually the dominated convergence theorem does not apply for the integral of [1-e^(it) ]/t^2 over [ \varepsilon, R], as \varepsilon ->0. The same goes for the integral over [-R, -\varepsilon]. The problem is that the imaginary part of the integrand given by -sin(t)/t^2 is not integrable in a nbh of 0. However, the dominated convergence does apply for the real-part of the integral, so I would suggest that you take the real-part first and then the limit in \varepsilon->0+.

[dolev]

I wrote some JUNK on my last math test.
Long story short, let's say my teacher wasn't impressed.

[Vienticus]

3:55 You erased a white board with your fingers and I wanted to scream at my monitor.

[youtuberdisguiser]

Compkex analysis also known as Funktionetheorie in german <3.
Btw i reakly gotta thank you. We had this but we werent given enough examples for my slow brain to comprehend it.

[tomatrix]

Wow I love compex analysis so much. It’s a shame they don’t do this for engineers, in my case atlteast

[lucapeliti]

It is simpler to set (1-exp(i and then use the theorem of residues.

[timurpryadilin]

according to the same calculation, if we replace cos with sin, we get 0. This is nonsense because we have a singularity at x=0. I think it would be better to show that your function is well-defined around 0 using the L'H, for example.

[GhostyOcean]

I hope in grad school I can do some cool work in complex analysis. I already took the basic course in my undergrad and want to see what I missed out on.

[perappelgren]

Big and small "circle" @ 0:59. Growing up, those used to have constant radii. 😂😂😂

[tretyakov]

I separated this integral in two single integrals, calculated integral of some beautiful exponent using Euler Puasson integral, find Im of all that and got sqrt(pi/2). Help me understand where is the mistake? May be we can’t use Euler Puasson integral with complex alfa?