Cauchy Residue Theorem, Introduction

In this video, Will from Stem Cell showed us how to integrate the integral of 1/z over a closed contour. (yes, he will also talked about the Cauchy Residue Theorem) #CauchyResidueTheorem

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blackpenredpen | 曹老師

[stemcell]

Big thanks to blackpenredpen for hosting my video! My channel features a bunch of interesting topics in maths and physics. I'm still finding my feet right now, but I hope to have some video series that cover mathematical topics in detail (group theory, complex analysis, ...) as well as some curiosities (pythagoras's theorem proofs, the hidden link between electricity and magnetism, ...)

[MattMcIrvin]

Cauchy's Residue Theorem, and the techniques that make use of it, are the closest thing to magic that I know of in mathematics.

[rybaplcaki]

Pls more Complex Analysis visual way!!! That thing is so rare on yt...

[TheRandomFool]

I don't recommend using i as an index of a series when dealing with complex numbers. It can create confusion.

[nicolasrios]

I love that what he is doing is going from an integral, to essentially the definition of an integral. I love it.

[jacobharris]

Thank you very much for explaining the residue theorem in a intuitive manner. This is my first exposure to the subject and it's making me excited to take complex analysis, whenever I get to it. I can't wait to start solving some integrals with this!

[ianmoseley]

Would like a bit more explanation about residues

[anweshaguha]

I have a list of things I wanna do when I grow up and start earning. There are 2 YouTube channels that are on that list of which I want to become a patron of : Blackpenredpen and Kurzgesagt.

[angelmendez-rivera]

To understand how the complex numbers C form an inner product vector space, it suffices to see that a + bi = (a, b). Then we can defined the complex product in terms of matrix-vector products. Let M = [(1, 0); (0, -1)] and N = [(0, 1); (1, 0)], the permutation matrix. Then the complex number product between complex numbers x = (a, b) and y = (c, d) is given by the vector xy = (a’Mb, a’Nb), with a’Mb and a’Nb representing nonstandard (non-Cartesian inner products).

[bernarddoherty]

Took a little bit of time not to be distracted by the accent. I am used to listening to that accent giving an excellent lecture on ancient Briton or the flora and fauna of the Amazon rain forest. Lol. But, it was an excellent demo. BPRP, do you own version please. I could listen to you all day! You should do a little stand up comedy on the side!

[Higgsinophysics]

Just as I am learning about this in class. Thanks for sharing it was very clear!

[BALAKRISHNADAMARLA]

Excellent explanation.
One doubt
How 1/A2 is becomes 1/A e^I psi (Apsi e^ipi/2 psi

[alexyuri_]

I was just studying residues lately... It is not so simple to fully teach these things in one video because imagine that you have an essential singularity, then you must expand the function in a laurent series to find the residue. Or if you have a pole of order 4 for example... Complex analysis is quite the endeavor

[PunmasterSTP]

Cauchy residue theorem? More like “Cool, I’m glad to hear about ‘em”…your explanations that is. Thanks so much for sharing!

[carmelpule]

I prefer to look at this in another manner,
dz/z the numerator dz rotates at exactly the same rate as 1/z but in the opposite direction so dz/z does not rotate and the integral is linear adding up to 2pi for a circle circumference. For other situations where the product F(z) dz, rotates, the summation is zero. most of the time,
It is like a person going out for a walk if he changes direction to go back to the same spot he started then the integral is zero, but if he does not change direction then he covers some distances in his integral.

[jrleighton]

This is fascinating. My favourite mathematical problem is below. And it relates, I seem to think, to this video explanation of Cauchy's Residue Theorem. But, for the life of me, I cannot understand why my problem has such a 'nice' answer, which it does. You can solve it geometrically (Desmos will get you a quick answer) or algebraically, more satisfying, but you need to know L'Hospital's Rule. Here it is: imagine an n-sided regular polygon with an incircle and a circumcircle. Let's define A as the area between the circumcircle and the polygon, and B as the area between the polygon and the incircle. What is the ratio of A to B as n tends to infinity? I'd love it if someone can enlighten me as to why this problem has the 'nice answer' that it does. Many thanks

[rajendrasinghrathore]

Your videos are awesome
I have a doubt which i have been trying for it about 1 week but not getting it. Question is related to trigonometry
Q) I will be very thankful if u or anybody could help me out!
Thanks in advance

[timotejbernat]

The summation index i is easily confused for the imaginary number i, I’m sure I’m not the only one who was caught off guard by that several times

[N_CommanderShepard]

Ahhh good memories of complex analysis. Do some partial differential equations!

[arequina]

I just subscribed. Brings back memories.