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

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

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