Polar Graphs and the Complex Derivative

[@juliavixen176]

I've just got to say that this is the best explanation of the complex derivative that I have ever seen.

[@imaginaryangle]

Thank you!

[@ShadeTDW]

Something you could look into would be Geometric Algebra, seeing as it allows you to divide vectors, would be cool to see if everything still works out the same. These videos have all been extremely interesting, and I can't wait to see what topic you will cover next. Notifications on for sure!

[@imaginaryangle]

Thank you for turning notifications on! There's an amazing video by Freya Holmér that digs into the connection between vectors-as-complex-numbers and Geometric Algebra (though it focuses on multiplication), I recommend it! kzbin.info/www/bejne/nqW8mWCKppyNeKssi=_TBtMdBa2hlZTDB8

[@ValkyRiver]

@@imaginaryangle There is also a video about Geometric Algebra by sudgylacmoe kzbin.info/www/bejne/bGHdkJumeqanepo

[@ValkyRiver]

@@imaginaryangle There are also many videos about Geometric Algebra by sudgylacmoe

[@imaginaryangle]

Thanks! I will check it out.

[@jedglickstein]

Your videos are getting better and better. What I like most is your ability to describe familiar concepts (here, the derivative as a ratio of input to output) in unfamiliar ways. It feels like taking the ideas I first learned in high school and slightly tweaking an “abstraction” knob. Fitting, given the beautiful images at the end of this video!

[@imaginaryangle]

Thank you! The way I thought about things in high school and the questions I had back then inspire a lot of these videos.

[@tiagobmx_st6247]

I am just learning the derivative concept, and seeing this with complex numbers is just astounding. Also great explanation, I could understand pretty much everything even though my math level is below this topic, which makes me way more curious. Your content deserves to be at the top

[@imaginaryangle]

I'm happy to hear that, it was my intention to make it accessible!

[@oreo_smoothie74]

0:30 "and along the way, learn how to *extend* the concept of derivatives to complex valued function" That was nice

[@gregwochlik9233]

Absolutely beautiful.

[@imaginaryangle]

Thank you very much!

[@Filup]

I took a complex analysis course a few years ago, so my recollection of it is somewhat limited now. Videos like this really make me stop to think about about questions I don't even really know how to ask, such as questions that involve exploring non-analytical cases with singularities. I love these videos, and they are a massive breath of fresh air to get my mind off my studies and onto something else, if only for a few minutes.

[@imaginaryangle]

Thank you! If you're not familiar with Math3D, it allows you to parametrize and draw complex functions, and it understands complex numbers directly, so it's much easier to iterate and experiment with different expressions and edge cases. There's a link in the description with the graph from the video as an example you can start from.

[@WhattheHectogon]

A truly remarkable video...thank you so much for the education and entertainment!

[@imaginaryangle]

You're welcome, and I appreciate the kind words!

[@realcygnus]

Nifty AF !

[@imaginaryangle]

😊

[@guigazalu]

20:50 Finally, imaginary angles!

[@imaginaryangle]

Had to sneak them in somewhere 😉

[@andrewporter1868]

Cool stuff, but something I've come across is partial derivatives and whatnot, and the obvious fact that although mapping from complexes to complexes may appear to require 4D plotting, on the contrary, every x + iy for reals x and y defines a plane, but the conjunction of two planes orthogonal to each other with one of their axes collinear suffices to define an infinite Euclidean 3-space just as the conjunction of two sets A*B forms a plane. Letting C denote the complexes, our Euclidean 3-space formed from C*C would be visualized by mapping all X + iy to f(X+iy), and all x + iY to f(x + iY) where X and Y are held constant as x and y vary over time. Wolfram Alpha just visualizes complex plots by showing the 3D plot of the real and imaginary parts separately since at that point, you have only 3 variables. Alternatively, perhaps it might be shown in 3-space two planes parallel to each other, and the path that each point follows according to the rules of the particular function over the complexes, so for example, f(z) = z would, for some finite rectangular neighborhood of input points, just show a rectangular volume; f(z) = z^2 would probably show for real t, |z|^(t+1) e^(i(t+1) arg(z)) as t increases linearly with time which would show some kind of cylindrical solid when Re(z)^2 + Im(z)^2 is less than or equal to 1 with funny transitional stuff in the neighborhood of 1, but a sort of hyperboloid volume outside that; etc. Sorry, I went on a tangent there (pun not intended). Anyways, is there a definition or analog of the derivative for 3-space for the intuitive idea of the plane tangent to a point on a surface?

[@imaginaryangle]

There's a lot of different ways to add "virtual dimensions" (color, time etc), or compress existing ones, and getting creative with the mappings gets really rewarding :) About your question, there is indeed this analogy with a plane tangent to a surface at a point in 3d space. The last graphic that appears in the video is also implemented in Math3D, there's a link in the description, so you could experiment there to find out how that works. In this case, you have a constant vector following the imaginary component of input mapped to the 3rd dimension (like "1" is mapped to the horizontal axis when doing regular R -> R graphs), so you'd need to combine the derivative with that vector to define the tangent plane. Two vectors and a point uniquely define a plane.

[@Jaylooker]

Polar coordinates of a parameterized space are very similar to a Fourier transform.

[@masteroftheinternetverse1296]

I had an interesting idea. What would happen if you took Pascal's Triangle and made it 1 face of a square base pyramid, and made the other 3 sides the equivalents of the triangle but for i, -1, and -i? What would happen on the inside?

[@imaginaryangle]

I don't know, it depends on the rules you come up with for working out the inside; you need to have a generator function for values "behind" the values on the face that agrees with what's given by the corresponding generator function coming in from the opposing face. I would suggest starting by defining a discrete 3D grid each number lives in to make the collection of arguments for the generators straightforward, and then experiment with different rules. A hint about how to come up with extensions is in this video: kzbin.info/www/bejne/p2PHkqSDh5ifaJY

[@angeldude101]

"Even if everything else somehow made sense, we still could not divide vectors." That's where you're wrong kiddo! 😎 It wouldn't make a huge difference though, since the result is effectively a Complex number anyway (assuming the vectors are 2D), which is distinct from a 2D "vector" in this system despite having the same components.

[@imaginaryangle]

Not conventionally without extra definitions, and this was not the place to go on a huge tangent 😁 But good catch!

[@guigazalu]

Yeah, what is it about not dividing vectors? I am always dividing J₃ vectors, and it is very much possible! They keep exploding, though.

[@Yusuketh443]

hi :3 UwU