Introduction to Complex Numbers - Complex Analysis #1

Introducing the complex numbers and complex analysis. This is the first video in a series covering the topic of complex analysis. We begin by introducing a complex number. Then we investigate the effects of multiplying any number by the imaginary number i. Finally, we take a look at some of the visualisation tools that we will use in later videos; phase portraits and modular surfaces. Please subscribe!

My aim for this series is to introduce complex analysis in a visually intuitive manner, and we will be starting with the very basics of complex numbers. We will start visualizing some simple functions very early in the series. Don't expect much in the way of rigorous proofs and definitions, there are plenty of text books for that. Instead, this series will aim to give you some visual intuition, that I hope will make any future study easier and more enjoyable.

In this video:
00:00 Introduction
00:30 A complex number
02:17 The imaginary number "i"
03:53 Visualising a complex number
05:49 Multiplying a number by i
06:36 Powers of i
08:53 Introducing complex analysis
10:43 Visualisation tools - phase portraits
12:42 3D phase portraits (modular surfaces)
13:42 cos(z) and cosh(z)

In this series:
4 - [Coming Soon] Multiplication of Complex Numbers and Functions
5 - [Coming Soon] Division of Complex Numbers and Functions
6 - [Coming Soon] Complex Differentiation and Analytic Functions

Extra Visuals (No Commentary):

Credits:

[TheMathemagiciansGuild]

I'm pretty new to making educational videos, so feedback is welcome.

[ricardasist]

Keep it up! These videos are incredible, the potential reach of these videos is immense

[NonTwinBrothers]

Out for real this time! :D
Great stuff as always!

[mnada]

Great video, thank you. I am looking forward for complex integration.

[elibennett]

I love your videos so much. Please keep making them. I learn so much! You are such a great teacher.

[thefractalistic]

This is genius! I've been struggling to make a mandelbrot set by hand, but this video surely helps alot!

[jackcallaghan]

Very nice visualisations! It would be interesting if when you model the 3D geometry of the function that there was a projection onto the plane of some sort for reference. For example in Section "3D phase portraits" if you used a collimated light source from above the object, the shadow would be an accurate projection. Or alternatively map the image of the phase portrait onto the plane.

[jexyl]

Loving the visuals! Wouldn't the rotation @13:32 be in the opposite direction? Just a sanity check since I'm still new to mappings

[abhishekpanthi]

13:42 wow...it represnts soo many graphs of cos(z) when seen with different angles that's really interesting to know

[SuperDeadparrot]

If you took a regular equation like x = vt and replaced t with s*exp( iu ) what would that mean?

[Gekko-ti]

can I get a 3d object file for the 13:50 cosz figure so that I can resin 3d print it?

[ermenleu]

didn't realize there were complex numbers down under 🤔

[pocojoyo]

2:17 Im not so sure i "is always defined as the positive square root of -1". Actually, i is defined better by i^2 = -1

[thomasolson]

[x, -y]^n= (x+i*y)^n
[y, x]
Now go crazy. Don't limit yourself to two dimensions with that function. sqrt(y^2+z^2). Just think about that... A 2D-rotation matrix in R^n space. Easier to learn than quaternions and you are not limited to R^4.

[krissantos]

🤓.. yeah, you lost me the variable named... i 😃