Complex analysis: Holomorphic functions

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This lecture is part of an online undergraduate course on complex analysis.

We define holomorphic (complex differentiable) functions, and discuss their basic properties, in particular the Cauchy-Riemann equations.

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I tracked down the origin of the word 'holomorphic'. It turns out that Briot/Bouquet (two students of Cauchy) introduced it in the 1875 second edition of their book about elliptic functions. The previous edition of their book had adopted Cauchy's term synectic, but in the second edition they wanted to focus more on meromorphic functions (which didn't previously have a name). Their stated explanation for the names is that a holomorphic function "resembles an entire function" (whole) in a particular region of the complex plane, while a meromorphic function "resembles a rational fraction [of entire functions]" (part) in a particular region of the complex plane. That is, the "-morphic" part is intended to mean "resembles" or "is shaped like", so that "holomorphic function" should read as "whole-type function" or "resembling a whole function", rather than "whole form". Your confusion about this inspired me to add the relevant clarification to Wikipedia, whence it will hopefully percolate into modern sources. (The original explanation is repeated in multiple 19th century sources I found, whose authors directly looked at Briot/Bouquet's book, but seems to have been lost in recent sources.)

jacobolus

I like where this is going, can't wait to see your take on contour integration, Cauchy's theorem and so on. Thank you for another great lecture!

leandrocarg

when i was first learning this a few months back, the cauchy riemann equations were taught completely differently. However when you started off with the definition of real differentiability being whether or not you are able to approximate the function linearly made them feel so natural, like they just pop out, it's so nice

jakubszczesnowicz

The best thing about Sir Richards is he knows these things in very depth and for him these topics are simple but he still teaches with such enthusiasm and excitement.

vaibhavshukla

It's fascinating that the only reason why complex functions are infinitely differentiable if they're differentiable once, unlike differentiable functions from R^2 to R^2, is due entirely to complex multiplication. That's it. It gives us the CR-equations, and then everything else follows.

f-th

Looking forward to the rest of the lecture series.

FisicoNuclearCuantico

wow, when I learned the notation df/dx never bothered to question it, and because of that it seemed some properties were just thrown there, but with a lim makes so much sense, thanks for sharing so much :)

arturjorge

I thought the reason it was called holomorphic was because it was comparable to holographic

codatheseus

In general, satisfying the Cauchy-Riemann equations is not sufficient for complex differentiability. If the derivatives of the real and imaginary parts are continuous, however, then it should be the case.

MartinPuskin

Maybe it is called holomorphic - "holo" -> entire and "morph" -> shape i.e the shape remains entire after the transformation by such functions - which is kind of true as these functions locally preserve angles when their derivatives does not vanish.

tommike

Thank you so much for this fabulous lecture! And also, if possible, I would very much like to see "topological or integration free" proof of the the fact that "derivative of holomorphic fuction is continous". Ahlfors said at the begining of chapter 4 in his book that such a proof has been found, but doen't present it in the book. And I also don't see such a proof in other complex analysis textbook. Thank you so much again!

ycchen

This is my favorite math channel to listen to in headphones while at work. Richard Brocherds is a great expositor, and unlike a lot of lecturers does a great job speaking mathematics, and making the ideas on the board audible. It is difficult (at least for me) to learn mathematics "by ear", but I am very grateful for these videos as impressively clear and followable examples of mathematics teaching in the medium of audio lectures (a technique introduced by Vivienne Malone Mayes, one of the first Black woman math PHDs).

abstractnonsense

Great lecture, is a pleasure to hear a Professor talking about Complex Analysis and address the fact that a Holomorphic function is not exactly equal to Analytic function. Great Youtube channel.

Homi

I think the examples at the very end of the video are not correct. For instance, x and y are certainly not holomorphic as they are Re(z) and Im(z).

Some examples there could be z or z^2 = x^2-y^2+2xyi, ...

Am I right? or I'm missing something??

unalcachofa

3.10 is an epiphany to me. Never did I get it, but learned to use it.
In the case of a real variable I still dont understand how it makes sense to split up the symbol df/dx when using, say, 'integration by parts'. Would the limit part follow the bottom half of df/dx around for it to make sense?

filipjohansen