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

Video Description

Unlock new career opportunities and become data fluent today! Use my link https://bit.ly/MathemaniacDCJan22 and check out the first chapter of any DataCamp course for FREE! 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📖📖 (Simply connected version) https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem (Homologous to 0 version) https://www-users.cse.umn.edu/~brubaker/docs/8701-F13/dixon.pdf Surely, You’re Joking, Mr Feynman: https://www.amazon.co.uk/Surely-Youre-Joking-Feynman-Adventures/dp/009917331X Math StackExchange thread: https://math.stackexchange.com/questions/253910/the-integral-that-stumped-feynman Residue theorem example links: Wikipedia: https://en.wikipedia.org/wiki/Residue_theorem#An_integral_along_the_real_axis [The example that I nicked from] You will need - estimation lemma (a.k.a. ML lemma): https://en.wikipedia.org/wiki/Estimation_lemma 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: Classifying singularities: https://sites.oxy.edu/ron/math/312/14/ws/24.pdf (page 1) More ways of calculating residues: https://en.wikipedia.org/wiki/Residue_(complex_analysis)#Calculating_residues YouTube videos talking about exactly the same integral (though they all assume quite a bit of familiarity of the above): Flammable maths: https://www.youtube.com/watch?v=YWBdwYr6PGg qncubed3: https://www.youtube.com/watch?v=DvND8J32gOA tahiri math science online teaching: https://www.youtube.com/watch?v=oGEo_BxqZ3c 🎶🎶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: https://forms.gle/QJ29hocF9uQAyZyH6 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: Facebook: https://www.facebook.com/mathemaniacyt Instagram: https://www.instagram.com/_mathemaniac_/ Twitter: https://twitter.com/mathemaniacyt Patreon: https://www.patreon.com/mathemaniac (support if you want to and can afford to!) Merch: https://mathemaniac.myspreadshop.co.uk For my contact email, check my About page on a PC. See you next time!