Complex Analysis 26 | Keyhole contour

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This is my video series about Complex Analysis. I hope that it will help everyone who wants to learn about complex derivatives, curve integrals, and the residue theorem. Complex Analysis has a lof applications in other parts of mathematics and in physics. In particular, we can solve real integrals by going to the complex realm.
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[@idk-vj8tz]

Great work! This helps one see beyond textbooks.

[@brightsideofmaths]

Glad you think so! :) And thank you for your support!

[@Ghetto_Bird]

Man you are such a legend, as crystal clear as always!

[@brightsideofmaths]

Thanks! I appreciate that! :) And thanks for the support!

[@synaestheziac]

Can’t wait for Cauchy’s formula!

[@souravdutta8630]

Superb understanding sir

[@youteubakount4449]

Amazing series, thank you!

[@brightsideofmaths]

Glad you enjoy it!

[@Ren-wi1gf]

Thank you so much for having these series m(__)m I wouldn't otherwise have such a good master explain these concepts while trying to self study

[@individuoenigmatico1990]

The proof works exactly the same if g is holomorphic in a disk D-{z0} where z0 is not necessarily the center of the disk, am I right?

[@skillick]

Why are we justified in taking the limit delta --> 0? I understand the result flows nicely when we do, but are we indebted to the limit we took somehow? Thanks for your videos!

[@brightsideofmaths]

What do you mean by "justified" here? The limit exists and we can do the calculations. Do you prefer not taking the limit but doing it another way?

[@skillick]

@@brightsideofmaths I mean in the sense of why is it useful to see what happens to this shape in the limit if we are trying to say something about the shape itself? I appreciate your response, I think I have a deeper misunderstanding I need to address before moving on!

[@brightsideofmaths]

@@skillick The delta was a helping variable from the beginning. So we want to get rid of it in the end.

[@skillick]

@@brightsideofmaths Thank you!! That caused the penny to drop for me!

[@kayebennett7867]

Why do we have to consider some non-zero delta and take the limit? Is not possible to choose directly delta equal to zero at the beginning?

[@brightsideofmaths]

With a zero delta, we cannot apply Cauchy's theorem as we have learnt in the former videos. I wanted to show a straightforward proof of this important fact :)

[@kayebennett7867]

@@brightsideofmaths But why exactly we cannot use the version of the previous part? If delta is zero, we are still able to construct the polygon paths, right?

[@brightsideofmaths]

@@kayebennett7867 Yeah but the region is not the correct one then.

[@paul_tee]

Whoa is this a face reveal

[@brightsideofmaths]

It's a least my face :D