Complex Analysis 6 | Cauchy-Riemann Equations

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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.

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00:00 Intro
00:16 Two notions of differentiability
06:39 When is the vector-matrix multiplication a complex multiplication?
09:04 Deriving the Cauchy-Riemann equations

#ComplexAnalysis
#Analysis
#Calculus
#Mathematics
#curveintegral
#integration

(This explanation fits to lectures for students in their first or second year of study: Mathematics, Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[brightsideofmaths]

I didn't write it down again but of course, in the last statement, you still need the total differentiability of f_R to get the equivalence. This is something you should not forget :)

[jms]

This is so much more elegant than the way I learned in in a mathematics-for-physicists course (where you just demand that f'(z0) has to be the same whether you approach z0 parallel to the x-axis or the y-axis). Very nice series so far!

[retiredmeme]

Really liked how you briefly summarized the concepts from the previous videos and slowly built up the theorem in such an understandable way, sets a new standard for teaching mathematics.

[jaimelima]

I love the approach relying strongly on normed vector spaces. It is so clean and modern. Hopefully there will be space to talk about C^n -> C^n in the future and how C -> C generalizes or not in that case. Thanks!

[NewDeal]

00:00 Intro
00:16 Two notions of differentiability
6:39 When we can use vector-matrix multiplication?
9:04 Deriving the Cauchy-Riemann equations

[ishaanivaturi]

I'm currently taking a class on complex variables, but it is not a proof based one which results in formulations like this being thrown around seemingly out of nowhere. As a student who took real analysis I was struggling a lot because I just wanted to understand the subject. This series is proving incredibly helpful, and I am finally able to get it.

[jongxina]

This is pretty elegant and I like the connection with the total derivative in R2.

[mertunsal]

it would be awesome if we could have a video every day :))

[l.s.]

Good video! I really wanted to understand the Cauchy-Riemann equations but struggled with it. However, it felt almost trivial for me in the video!

[NewDeal]

I assume it'd be cool if you'd reference it when you get to that subject. Coupled with your lucid explanation of all the lemmas and proofs, some visual intuition is exactly what's needed to create a killer package for an all time best complex analysis course that Youtube'd ever seen)

[NewDeal]

The content is on top as always, but haven't you though about adding timestamps to your videos? When you deal with those longer than 10 minutes you may easily forget something important and havin a timestamp to quickly revive the idea could save a lot of time and make all the courses easy to navigate.

[bhatiyasagardevanand]

Your work is great sir it very helpful for me💯💯 (engineering students)

[vazn]

I don't understand the point about complex multiplication.

You said that "if we want c-differentiability, we need the jacobian to have this form" implied the jacobian should represent complex multiplication of two complex numbers.

But why is it enough at all ? We can imagine complicated complex functions that cannot be represented as multiplications, yet still be differentiable (or maybe not ?)

What i understand is that we proved this fact:

If the jacobian represent complex multiplication (i omit all technical details), then f is c differentiable.

But why should the converse be true at all i don't understand ?

Great content anyways, I'll proceed and admit this result for now x)

[igorinoue]

There is something that is not clear to me. I understood that if f is differentiable then the Cauchy-Riemann equations must be satisfied, but I don’t understand why the converse is true.

[aarian]

Very nice explanation. which software is used for these presentation?

[frankansari]

This is really interesting. Because for real function if the function has no gaps or sharp edges it is always differentiable. But for complex functions as it turns out this is not the case. A simple counterexample is u(x, y) = x with v(x, y) = -y.

[adammac.]

Are you following a textbook when you do these videos? i.e. how do you decide what to cover?

[fouadio]

Hey there ! I really wanted to start learning Complex analysis during the summer and really just couldn't settle down to actually read the books and watch the videos. But now, I am able to ! Despite being busy with college and those pesky finals, the fact that those videos are in bite-sized pieces really helps a lot in getting introduced to concepts in complex analysis. Thank you ! :)

[nucreation]

Hello. Thanks for another great video! I have a question; would you mind helping me understand? From what I gather, if f is complex differentiable at a+ib, then it implies that g is totally differentiable at (a, b) where g((x, y)) := [Re{f(x+iy)}, Im{f(x+iy)}], but it's not the case the other way around right (ie g totally differentiable at (a, b) implies f complex differentiable at a+ib)? Also, is there an easier way to tell if a multivariable function is totally differentiable at a point? For example, will a function be totally differentiable at a point if all of its partial derivatives exist at that point? Thanks again for all of your hard work and clear explanations.

[shuvohasan]

Your channel is a wonderland, I'll use my brain...💖💖💖