Complex Analysis L06: Analytic Functions and Cauchy-Riemann Conditions 

Thanks for providing the necessary context. For Showing us non analytic functions also and putting a meaning around how rare analytic functions are.

These videos have been awesome, so far. Your approach to the topic is a little different from previous classes I've taken, and this is actually perfect because it's helping me understand the ideas in more nuanced ways than I had before. Thanks so much!

Fantastic series! needed to brush up on this for a mechanics book I'm working through.

I don't recall ever seeing it in 25 years. It'd be nice to see the complex derivatives derived from the polar form. Express the Cauchy-Riemann conditions in polar form, or find an alternative set of necessary conditions that are more easy to express in said form. And, then come back to monomials.With the hope that infinite differentiability of analytic functions is easier to intuit in that form.

Great presentation thank for teaching us.

16:29: 2(1/2), it is like platform 9(3/4)...😅😍, it is as like as this sentence: "your teaching style is like magic".

You are great at these videos, you are really helping me improve my calculus. And i am at highschool its really hard to find stuffs that i can understand with my limited knowledge... so i wanted thank you ❤

wow! this is a gem of a video

You gives your best in teaching, awesome.. thank you sir!

Thank you so much for this great lecture and for including your own mistakes along the way. I struggled in math in high school and so gave up on math/science as soon as I was able to, and only decided to try again in my late twenties; after a few years of playing catch-up, I just started a degree in engineering. I spent so long thinking I was just "not a math person" that even having accrued plenty of evidence that I can in fact learn math if I try, it's still easy to assume everyone fundamentally has it together except for me and I'm the only one falling into "stupid" holes all the time. 21:15: would love a primer on the concept of "measure" some time-- I first encountered it when I ended up teaching the lab for the precalculus catch-up class not too long after taking it myself, and found myself saying something similar about an example without thinking about it-- "if I randomly chose a function, it probably wouldn't be the right one... wait, huh... can I say that? what does 'probably' even mean in this context?"

Thank you. (Additional thanks for including example at end of presentation.)

Thank you very much. It's very simple tutorial but very hard branch of math.

Wow! This is GREAT! This is a whole new world for me! 😂 ❤

great lecture

1) The principal logarithm function is discontinuous on the entire negative real axis, not just on z=0 hence it is not analytic there. But it is analytic everywhere else. 2) A complex function is analytic if and only if it is R²-differentiable and its partial derivatives verify the Cauchy-Riemann equations.

Couldn't find L04, and L05.....it would be great if they are uploaded. Anyways, thanks a lottttttttttttttttttttttttttttttttttt.

can someone explain why z^2 is single valued @2:52

from my own evaluation of the cauchy reimann equatoin i got [u(∆x, ∆y) - iv(∆x, ∆y) ]/ (∆x + i∆y) is this the same with what you got?

Fantastic! My interest in complex analysis was piqued by Roger Penrose’s road to reality, but unfortunately his writing is just too impenetrable for me. You present it in a far more easy to understand way.

25:00 why expand in a first order taylor series? Wouldn't that ignore higher terms that could affect the result?

Thanks

At 25:35 When you say "you ca pause and this is what you get..." This is really really not evident for me. What mean to expand something out in a first order serie?

I thought the complex logarithm wasn't analytic on the negative real numbers? It's at least not continuous there. So shouldn't we need to remove (-inf, 0] from C?

i'm guessing D must be open?

how does z/2 have two solutions. isn't it a one-one function

amazing video thankyou : lol it was stupid mistake

A question/suggestion: why do U treat e^z analytic but z^n (only for n=1,2,..) analytic without 0? A practical question in order to reveal some Global Idiocy in physics just by using a complex plain and contour integral as INDEPENDENT from the "path" in such a "perfect/divine" plane - there are just other just real number analytic and geometric solutions but this one is best simple (for the path in Minkowskian space-time)

19:40 z conjugate is not cuspy at all? it is equivalent to f(x,y) = x - yi which has real part x and imaginary part -y which is not at all cuspy. what do you mean?

I bet prof.Brunton lefthanded.

It's not the principle of the thing, it's who your pals are. And Riemmann is not your friend, he's some American who got his name mangled at Riker's Island.

how is Z^2 a single valued? although Z^2 can be from +Z or -Z?