Complex integration, Cauchy and residue theorems | Essence of Complex Analysis #6

I can't pronounce "parametrisation" lol

A crash course in complex analysis - basically everything leading up to the Residue theorem. This is a more intuitive explanation of complex integration using Pólya vector field. As is the case for all videos in the series, this is from Tristan Needham's book "Visual Complex Analysis".

You might notice that my explanation on parametrisation is a bit similar to the Jacobian, and you will be right! Jacobian is really important in this area (and also understanding complex differentiation and Cauchy-Riemann equations).

I have made this slower in comparison with some of my other videos, because when I myself watched some of my other videos that are faster, I couldn't comprehend if I was not paying too much attention on the screen, let alone the audience watching it for the first time. If somehow, miraculously, you think this is way too slow, feel free to speed it up!

I said the more general Cauchy integral formula is related, because in my original plan, I did want to say that Laurent coefficients take on exactly the same form, but it just occurred while I was finally editing the video that we don't find Laurent coefficients using integrals, and I don't want to send my Cauchy integral formula bit to waste, so here it is.

Throughout this video series, of course I have left out lots of theorems in complex analysis, only talking about the things that I find more "applicable" (read: more audience want to watch). Things like Fundamental theorem of algebra, or maximum modulus principle, or even winding numbers are not presented, but in my defense, they are not really "essence of" anymore, because they use the concepts that we have developed in this series instead - like Cauchy integral formula as seen here.

This video was sponsored by DataCamp.

📖📖MORE READING📖📖

(Homologous to 0 version)

Residue theorem example links:

[The example that I nicked from]

If you want to watch other videos on the exact same integral instead (although I think the Wikipedia page is a more “elementary” way of finding residues), you might want to have a look at:

YouTube videos talking about exactly the same integral (though they all assume quite a bit of familiarity of the above):

🎶🎶Music used🎶🎶
Aakash Gandhi - Heavenly / Kiss the Sky / Lifting Dreams / White River
Asher Fulero - The Closing of Summer

Video chapter:
00:00 Complex integration (first try)
06:01 Pólya vector field
08:18 Complex integration (second try)
12:27 Cauchy's theorem
18:39 Integrating 1/z
22:28 Other powers of z
28:26 Cauchy integral formula
31:43 Residue theorem
36:14 But why?

Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:

If you want to know more interesting Mathematics, stay tuned for the next video!

SUBSCRIBE and see you in the next video!

If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!

Social media:

For my contact email, check my About page on a PC.

See you next time!

[mathemaniac]

[hopefully, this time, the pinned comment won't disappear like last video, cos for some reason YouTube decides that a similar comment is spam lol]


This is the end of this video series, although I wouldn't say that complex analysis will disappear on this channel forever, just that it will only have occasional appearance if I feel like it.

[DrTrefor]

Very impressed with this complex analysis series, well done!

[nosnibor]

I watched this out of nostalgia. I am a retired Electrical Engineer and back in 1975 I covered this work on my final year maths syllabus of the HND, at Manchester Polytechnic, UK. I still don't fully understand it ; maths is so beautiful. I think this is applied in advanced control systems.

[samyaksheersh]

Why on earth wouldn't anyone watch this till the end? It's such a beautiful result so elegantly presented! Keep up the good work!

[yinq]

Great as always!
I never saw the "Work + i Flux" interpretation before. It's really helpful, just like the area/mass interpretation in real integral.

[nezavipavc]

Physics student here, and I have to say that this has made my understanding of complex analysis *so* much better! You explained everything way more intuitively than how our professor presented the material, so thank you for making this video!

[MaxxTosh]

By far my favorite video covering complex analysis, ever. You put together nearly all the integration topics so well!

[evaxu]

40:08 Watched to the end! An Oxford math student here :)) I am so grateful that you made this series of video. I was previewing complex analysis during the summer and suffering until I discovered your essence of complex analysis. It made the subject much less daunting and helped me a lot during the term! It even cultivates in me a love for complex analysis! There are not a lot of intuitive videos on university math like what you did. I was so excited when I found your channel and I recommended these video to all of my friends after the term has started :) Genuinely thank you so much! ❤️

[nathanisbored]

i watched to the end, i learned a lot of what i wanted to learn in this episode!!

[johnchessant]

Amazing! Really appreciate the effort you put into these videos, of course I watched to the end :)
To add the details for the integral: The magnitude of the integrand exp(iz)/(z^2+1) on the semicircle of radius R is bounded by 1/(R^2-1) while the length of the semicircle is πR. So the integral on the semicircle is O(1/R), which goes to 0 as R -> infinity. The residue at z=i of the integrand is the limit of (z-i)exp(iz)/(z^2+1) as z -> i, which is exp(-1)/(2i). So the integral is 2πi times that, which is π/e. A lovely answer!

[nicholasbohlsen]

22:35 I love seeing the multipole fields of electrodynamics appear as the Polya vector fields of the inverse powers. Its just such a cute connection between the multipole expansion and laurent series

[ativjoshi]

This single video feels like a crash course on Complex Analysis

[LAICEPS]

I love everything related to Cauchy's mathematical topics. Now I love this Channel.

[abdullahalsakka]

You are an amazing creator and I don't know how to thank you enough. I am a physics engineering student and I have been struggling so long to understand the intuition behind complex integrations. I can do them but unfortunately by memorizing how to do them, I was blown when I saw this simple intuitive video and I really want to thank you deep from my heart. Also as a side note, I really liked how you explained the complex integration using physical quantities like work and flux. Keep up the great work and I hope soon enough many people find this channel and explore the fun and intuitive sides of mathematics.

[samuelmarquardt]

I love that the ads are between segments instead of annoying me while he's explaining

[ЕгорКут]

I am supposed to start studying complex analysis in nearly 1.5 years from now. But it's really nice to have some good intuitions before you fill in the details. Thanks for your video!

[vkv]

As a physics student....this blew my mind words to convey my regards..thankyou very much for your effort..I was smiling out of pleasure whole thile time while watching your explanation...😌😌😌

[daigakunobaku]

Your video is the best explanation of Cauchy's formula I've ever seen, and I've read this part in, like, three different textbooks. Please, continue with this series, it is damn good

[mellamofields]

Such an amazing video, possibly one of the best vid of complex integration outhere, great job!

Btw, is this inspirated on the book Visual Complex Analysys from Tristan?

[YourLocalCafe]

I am a highschooler so this went way over my head, but i must say that you are doing a great job at making these and you must keep at it!
Your quality and the quantity with that kind of quality are both spectacular.
Your channel is heavily underrated, but hey i am here!