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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.
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
🎶🎶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
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