Rouche's Theorem and the Fundamental Theorem of Algebra

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In this video, I prove Rouche's Theorem, which relates the zeros of a function f(z) and the sum of two functions f(z) and g(z), assuming that certain conditions are satisfied.

Following the proof, I apply Rouche's Theorem to an important example involving the proof of the Fundamental Theorem of Algebra. I'm sure that the pure math lovers who watch my videos will be excited to see this proof!

Questions/requests? Let me know in the comments!

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This Fundamental Theorem of Algebra proof is slick as heck! Great video, great series. Please do Real Analysis soon!

bugson
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I learned Rouche's theorem many years ago but I forgot about it, and this was a very clear and thorough explanation. Great job!

dcterr
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My favourite teacher is back with an outstanding video. Thankyou very much.

yudhisthir
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I was just googling something about Rouche's Theorem, but ended up here and got a nice proof of the Fundamental Theorem of Algebra as a nice bonus, thanks!

Alex_Deam
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Please do more complex analysis videos! I'm going to take the grad version of it next Fall 2020. I took Complex variables in spring 2019 and I really liked it. Videos are smooth and very easy to understand. Keep up the good work!

CHRISTIANZEGARRA
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When looking at complex analysis proofs I always think:" huh neat, true true. Very clear."
They are so *cool* and almost straight forward, I donno they just make sense to me... Contratray to some more difficult linear algebra proofs...:]

eulefranz
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Very helpful videos. Watched them all and I'm looking forward some more!
Thanks a lot!

JoelRTLCosta
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You know men, I am deciding to throw away all my books and wait for your videos....thanks a lot.

shamsulkhaknurulkhakov
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Please, make a video about evaluation of CONTOUR INTEGRATION IN THE PRESENCE OF BRANCH
CUTS. It is still unclear for me to distinguish between pole and singularity.

shamsulkhaknurulkhakov
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Can you make a video of the most simple proof of FTA that is based on the Green's theorem for circles and polynomial analyticity?

xjuhox
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It feels like someday you're going to talk about L-function and Modular Form.

hafizajiaziz
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pleas also mention books which you think best for complex analysis and group theory

Preparing_and_Testing
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Why proof is important if some theorems practically work.
Why do we need their proof?

mousaalsaeed
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