What do complex functions look like? | Essence of complex analysis #4

A compilation of plots of different complex functions, like adding and multiplying complex constants, exponentiation, the power function (including nth roots), and logarithm. Issues like branch cuts, branch points, and branches in general will also be discussed as the result of inability to construct the plots. Finally, we will do a 4D rotation (composed of two 4D reflections) to the typical Riemann surfaces pictures, and see that it should be the same as its inverse functions.

The video is going to be jam-packed with visuals and animations, so while it may sometimes be too quick, you can pause the video; or you can just simply appreciate the visuals, the plots, and the animations.

Some interesting plots are usually the vector plots, like for the power functions, we have different regions of flow. The formula of 2(n+1) when n is positive can be left as an exercise - it is not TOO difficult to see why, but it is not the focus of the video, or not the primary feature that I want to discuss; or for negative powers, we have dipole, quadrupole, and octupole, and in general multipole, which might be familiar to physicists, because in electromagnetism, we use multipole expansions to see the dominant effects of the electric field.

Watch the previous video to see what the 5 methods of visualisation I am referring to, and also watch the Problem of Apollonius video for the next video!

Video chapters:
00:00 Introduction
01:01 Adding constant
02:51 Multiplying constant
06:14 Exponentiation
09:47 Power function - integer powers
14:11 Power function - complex inversion
15:39 Power function - square root branches
20:37 Power function - Riemann surfaces
22:53 Logarithm
26:51 Logarithm - 4D rotation

Music used:
Recollections - Asher Fulero
Stinson - Reed Mathis
Beseeched - Asher Fulero
White River - Aakash Gandhi

Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:

If you want to know more interesting Mathematics, stay tuned for the next video!

SUBSCRIBE and see you in the next video!

If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!

Social media:

For my contact email, check my About page on a PC.

See you next time!

[mathemaniac]

As always, pause the video if necessary.

I realise that making a whole video series about complex analysis is a really monumental task - much, much, much more than what I expected - but don't worry, I will still make them *eventually*, just that (1) I need a lot more time, so the next video is not going to appear at least until mid / late Dec, and (2) I might have to sometimes switch up, i.e. only occasionally putting out videos on CA, and not necessarily all uploads would be about CA. This is a problem I found myself into, like the group theory series, that I feel like some people might not like CA all the time (or at least, will get bored after something like half a year), which is why I want to give a heads up that this might happen if I feel like it.

This video is much better viewed with a good degree of familiarity of the stuff mentioned at the start, so PLEASE watch those first; and of course, watch the video on Problem of Apollonius as well for the next video on Möbius maps.

[jill]

This is why I freaking love complex analysis

[MathPhysicsEngineering]

This time you outdid yourself, an absolutely incredible job! It gonna be on the must-watch list for my students. You set the bar really high this time, you inspire me to put more work into my videos!

[ProCoderIO]

LOVE complex analysis. These visual tactics to SHOW complex functions is pretty cool. Thanks!

[borial]

It's clear how much effort goes into these videos, thank you! Great work!

[battlelance]

Just discovered the channel today. I am a big fan of how you explain the topics in all your videos. I look forward to seeing what you make in the future :)

[johnchessant]

These animations are brilliant! This really makes clear what a branch point is and why it's necessary for sqrt and log. I can't wait to see the visualizations for the Mobius transformation!

[enbyarchmage]

Wonderful video! Little gems like this are one of the things that make me keep loving math 🥰

P.S - Try not to worry too much about your French pronunciation. My French teacher (who is French herself) once told me that "French words are like Christmas trees: full of seemingly ornamental letters".

[Iudicatio]

My favorite are the Re-Im plots even if they are less useful because looking at them I am in awe of the beauty and the complexity of the 4th dimensional world they came from. In the others it's not quite so obvious to see.

[yurisich]

Those 29 minutes went by fast. I can't believe you managed to get that much into one video. Looking forward to the Möbius maps video!

[kennethvaten]

Hey, I think there's a small mistake at 5:03. Matrix multiplication on the right side of the equation, the second row first column spot should be "ad + bc"!

[antoninperonnet]

The inverse function on the z-w plane is really nice and helpfull, thank's !

[gn]

What serendipity to have your next video be on moebius maps just when I will need it for uni - brilliant series man.

[hommorphism]

wow! amazing video! the apollonius video was the first of your videos i watched and i absolutely loved it! thanks for making this content. waiting for the next video :)

[PunmasterSTP]

Branch cut? More like “This is the stuff!” I’m enjoying your videos immensely and I can’t wait to watch the rest of them.

[angeldude]

This actually solves one of the mysteries behind a fractal that I found. I didn't know why there was a discontinuity along the real axis and now I know, because I'm taking roots of the inputs. I also managed to find a single white pixel at the origin which would be the branch point. The exact roots also depends on the inputs which demystified what frankly should've been obvious which was white around the point -1. The specific roots that I was taking was z+1. When z is -1, taking the reciprocal before using it as an exponent makes it blow up to infinity. Now I'm curious what I'd get if I tried plotting it in 3D so it would avoid the branch cut.

[tanchienhao]

Awesome channel and video!! Im glad somebody is finally making videos to illuminate the beauty of complex functions

[MichaelMaths_]

I like the z-w plane method the most, but the 3d plot showing different 4d perspectives is really amazing and mind-boggling when I first saw it!

[TheJara]

Brilliancy as your standard..so we thank you and get messssmerised by your presentation...

[MsSlash]

Can’t wait for the next video about Möbius Transformations!