What does a complex function look like? #SoME3

Join me as I explore the different ways we can visualize a complex function, to find which one deserves to be called their true graph.

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0:00 - Quick introduction
0:30 - Why can't we just plot a complex function?
1:45 - Mapping between 2 planes
2:35 - Grid mapping
3:07 - Reading a grid map
4:15 - The problem with grid mapping
5:35 - Colors to the rescue!
7:07 - Mapping hue and brightness
8:11 - Contour maps
8:43 - Domain coloring: z/(z^2 + 1)
10:07 - Domain coloring + contour lines
10:40 - Domain coloring: z^2
11:00 - Domain coloring: e^z
12:05 - Domain coloring: z^5 + z^2
12:30 - Domain coloring: tan(z) and (z-4i)/(z+4i)
13:20 - Going 3D
14:50 - |f(z)| + hue
16:20 - What is a graph?
18:12 - Projections and surfaces in 4D
18:53 - Graphing Re(f(z))
19:45 - Re(f(z)) + hue

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The bulk of the code I used can be found on github:

CC atributions:

Contour map of Mt St Helens

[TheTickterd]

I never fully grasped why four dimensions were needed for complex functions, but the way it was explained here finally made it click

[roygalaasen]

Love myself some #SoME3 in the evening.

[orlandojimenez]

I love how this video is 20 minutes long and it felt like 5 minutes. Everything was well explained and it just kept getting more and more interesting. Congrats!

[battleprof]

This has to be one of the clearest explanations from a SoME3 submission. Great work!

[leedavis]

I love your motivation at the beginning! You really hook the viewer with exactly what you're talking about, why it's interesting, and make promises for later in the video. It really kept me watching! In terms of clarity, I also loved how basic you started, letting anyone with even a small knowledge of imaginary numbers and what they are get by and understand the video. I think you could've done with maybe a short reminder of what imaginary numbers are and why were graphing their functions, but other than that it was fantastic. I also love your use of constant examples and animations, which really make sure the viewer is staying with you as your progress through the video. I think a couple of your steps could be more well thought through or explained, such as explaining polar coordinates, but even those were not bad and could be fixed with just a few seconds. In terms of originality and memorability, this was fantastic! I've never seen this covered before, but it's such an important problem with dealing with complex functions. I feel like most people take your first solution for granted when looking at the problem, so it's super interesting for someone to dive into possibilities we haven't considered for such a basic task.

[szymczakh]

thanks for putting this together. Wonderful explanation.

[RobertKing-ce]

only once did my teacher visually show the complex region on the white board, but this video helped it explain what it looks like. cool!

[miguelangelhernandezortiz]

I'm a mexican collegue student and i enjoyed the video, i always wanted to understand how to visulize a complex function and this is the first video that explain it well.

[HyperCubist]

It actually is possible plot a 4D graph, for instance a surface with two parameters, in a 4D space. The solution is exactly what we do with plotting 3D surfaces on a screen. We project it onto a flat screen, and then rotate around in three dimensions to examine different views. You also can take 4D information (the surface), project it to a 2D screen, and similarly rotate around in four dimensions, examining the surface from different points of view in 4D space. For instance one view might show the real output on an axis perpendicular to the xy plane (the z-axis), another view would show the imaginary output on a different perpendicular axis (the w-axis). And you can rotate partially between the two views. The difficulty is interpreting what your seeing, which is a matter practice, but it certainly is possible.

[gianlucadegliesposti]

I love domain coloring, I used in a paper to visualize complex numerical solutions to a differential equation and people loved it too :)

not only can you immediately see all poles/zeros/essential singularities/branch points immediately, but you also see the order of such just by seeing how many times the hue changes around a point

Mathematica has a beautiful color function to visualize contour lines and at the same time lines of constant real/imaginary parts that I found the most complete

[NewYokh]

Crazy interesting videos. Its gonna be so nostalig if future me sees this after having pursued a career of mathematics

[thiagoulart]

Literally the best video I've seen about this.

[pancito]

just one criticism: 16:31
you say that the graph of 𝑓(𝑥) is {(𝑥, 𝑓(𝑥)) ∀ 𝑥 ∈ ℝ} when in reality that only counts as the graph of a function 𝑓(𝑥) with domain ℝ, but not for any function whose domain doesn't span all real numbers.

for the rest, the video is an amazing learning tool, hope you the best!

[vladimirshitov]

Absolutely incredible video! Truly 3blue1brown level

[heatheretaithaha]

my colorblindness going wild on this one lmao

[ruudh.g.vantol]

A common 4th dimension is time, so the graph could also be a movie that dynamically shows how input leads to output.
Wave patterns are often shown like that, because the input is repetitive.

[John-xlbx]

Why doesn't anyone plot these like a vector field? Those plots often put little 2D vectors at points all over the domain. For many functions I think this provides a nice combination of accurate and intuitive.

[algea]

10:35 looks like 3 quarks forming into a baryon

[dAni-ikhv]

never could've imagined that by simply taking the sqrt(-1) we would ever get to 4 dimensions.

[PXWantonio]

You explained it perfectly, I understood Everything!