Complex Analysis L07: Analytic Functions Solve Laplace's Equation

This video shows that the real and imaginary parts of analytic complex functions solve Laplace's equation. These are known as harmonic functions.

@eigensteve on Twitter

[strippins]

These lectures make this material very easy to understand. I wish YouTube and content like this existed 20 years ago

[danielhoven]

I am truly riveted by this series. It's all stuff I've seen before, but I love how you can pull it all together in these lectures.

[amittksingh]

Thankyou professor for these lectures. I love how you connect everything in your lectures. and like pieces of puzzle that completes a picture it starts to make sense !

[deletedaccount]

Professor, I think the equation in 18:37 should be Ur = 1/R V theta (without the negative notation).

[rajendramisir]

Fascinating stuff! Excellent explanation and motivation.

[juniorcyans]

Thanks very much for explaining the notations! It’s very helpful for me to catch up the ideas.

[AlfredoMaranca]

I think many of us have seen this theory before, but I never got how beautiful it is, and more importantly, I never managed to like it as you make us do.

[florianvanbondoc]

18:59 thanks for explaining different notations, sometimes this is one of the biggest issues to understanding some math paper

[xiangwenyan]

It does take a few hours of mine to derive the polar system C.R.Conditions. And It's hard to imagine how could we approach a point with a fixed radius but only the infinitesimal angle is changed for somebody who graduated 10 years ago! LOL

[mosena]

Hi professor Brunton, thanks for doing these lecture series on YouTube. How can analytic functions be generalized to real-valued functions, and how can the contour integration be useful for real-valued functions?

[hyperduality]

Analytic convergence (syntropic) is dual to analytic divergence (entropic) -- Taylor series..
Analytic or predictable domains require the Taylor series to converge -- a syntropic process, complex differentiable or holomorphic.
Conformal maps are complex differentiable or holomorphic hence analytic and their Taylor series converges -- a syntropic process, teleological.
"Always two there are" -- Yoda.

[curtpiazza]

Wow! My head is spinning 😮 I'm
learning a lot ! 😊

[hoseinzahedifar]

The theorem of Goursat was amazing...🤩🤩🤩.

[annanor]

The polar derivation has defeated me for now, but perhaps I will come back to it after giving my brain time to grow...
8:29 but 0! = 1, n'est-ce pas?

[assadasdasdasdasable]

I chuckled a little every time Steve mentioned "hairy integral" xD

[SmithnWesson]

It's not just a log function that has levels.

The numbers themselves have levels.

exp(i t) = exp(i (t + 2 pi k))

for all integer k

It seems to me that any complex valued function also has levels.

Why is the Log function singled out as special? Is it because it has a singularity?

[hyperduality]

Rational, analytic functions are calculable or computable and hence predictable functions -- syntropic functions!
Analytic functions are syntropic or predictable -- algorithms or software programs.
Syntropy (predictable) is dual to entropy -- the 4th law of thermodynamics!
"Always two there are" -- Yoda.
Real is dual to imaginary -- complex numbers are dual.