Computing Definite Integrals using the Residue Theorem

In this video, I show how to evaluate definite integrals involving sines and cosines by taking advantage of the polar representation of complex numbers and then applying the Residue Theorem. I look at one simple example and one complicated example of an integration which takes advantage of the Residue Theorem.

In many ways, the Residue Theorem makes life a lot more simple by allowing the evaluation of integrals that would otherwise be difficult to compute using the techniques you learned from Calculus 2. I also hope that this video satisfies some of the requests I've been getting on this topic.

Questions/requests? Let me know in the comments!

[mikaelnuutila]

The residue theorem is the most badass thing ever.

[StarShootex]

I don't even know where to start but your videos are literally the stuff of dreams for students. Your simple straightforward 12 minute explanation trumps a 2 hour long lecture from my university professor.

Thanks for doing what you do :)

[stayawayfrommrrogers]

This channel is really underrated. Its definitely the best channel for analysis topics.

[Vyantri]

This playlist of complex variable lessons is literally a substantial contribution to humanity. As much as the invention of electricity, the wheel and the steam engine was.

[spasbanchev]

I've said this before but just to mention it again your videos are stunning. Very straightforward, easy to grasp explanations and because you go through everything quite quickly it makes it significantly easier to follow.

[leredsock]

The fact that there's 996 likes vs 9 dislikes... This video must be saving lives!

[duncanrichardson]

An excellent video and the presenters willingness to answer questions is very valuable. Thank you

[zekitopcu]

As a math major, I was ignorant of what complex numbers were for. I couldn't answer questions like "what does i even do?". After this theorem, I have solid answers.

[nataliemendez]

May God Bless you, your work and this channel.
Thank you!

[SadmanAhmedShanto]

You are great, brother. Good work on the series!

[sushilbarala]

Simple and precise explanation..time saving ..Amazing work 👌👌

[Odyssey]

Wow....honesly, this is keep making more videos

[sajidrizvi]

Damn. Thank you so much. I feel like I know so much more thank what I knew yesterday. Your videos have made my day

[glhell]

thank you so much! You are great at explaining

[muneebhasan]

1st lecture i got on residues that is to much

[ahmedshahabi]

I miss when math was only in real domain 😅

[willyou]

@ 8:30 could you have made a substitution u=2θ, du = 2dθ, and the limits would change from u=0 to u=2pi.


You still got that 1/2 factor, but without the assumption that its half the integral to 2pi. For sin(θ)^2n, its easy to guess, but not for other functions.

[grantmarshall]

Sorry if this is a dumb question...how does one determine the contour? Thank you

[rjbeatz]

Hello!
Can you recommend any books related to this topic?
Thank You

[zhongyuanchen]

I have a question. Don't we need the function to be analytic in order for us to use the residue theorem to evaluate integral? If we merely turn a real function into its complex representation, we will always get a analytic function?