Visualizing Complex-Valued Functions

[@whoisgliese]

Complex functions are one of the most beautiful things I've ever seen

[@Rinvids]

That transition to Part 2: Domain Coloring was smoother than Snoop Dog spreading butter on a hot slice of toast

[@raydson_S]

Wow! I can't even imagine any smoother example than this 😯

[@annevanderbijl3510]

Bro, you seriously did an amazing job at explaining. Well done!

[@lemmaxiom]

Thank you, glad you liked it!

[@deca0]

8:00 OH MY GOD WHAT A TRANSITION

[@TheJara123]

oh man I am speechless, among all the videos on KZbin on complex functions, yours is clear and mesmerising...please make more videos...and often...you are so good at it...

[@nipunviraj3185]

Recommended video for people who want to make an interest on mathematics. Great video.

[@_zweh7054]

I'm blown away by the amazing quality of this video, superb job

[@7ismersenne]

This is a beautifully made video. It has helped me understand a little more those mysterious (for me) complex numbers.

[@davidwright8432]

Very many thanks for a beautiful and instructive video! I got a great deal out of it, and will get the software you discussed. In half an hour careful watching I learned more about visualizing complex functions than I ever knew - even with some grad work in applied math! You also show how visually impoverished 2D, black-and-white or grayscale, graphs are. Like looking at a vast landscape thru a fixed pinhole.

[@Benjax_95]

Amazing job! Definitely one of the best videos I've watched in a while. The great graphs + the perfect music choice + your smooth and great explanations are simply all the perfect ingredients for this great work. Been trying to find an intuitive and good explanation of these weird looking graphs for a while, thank you so much!

[@lemmaxiom]

Thank *_you_* so much, I really appreciate that! It's awesome to hear that these videos are helpful, these kinds of comments are always motivating.

[@ep9196]

Great work here! I'll definitely be sharing this with others.

[@wu.gi.tai.]

Great materials, thank you so much for providing these. They are really helpful.

[@BoZhaoengineering]

I should've seen your video on complex function for the visualization. Thank you for making this very clear!

[@sigmaTrader3]

Very informative and thanks a lot

[@supu8599]

Every thing about this video was great

[@maxaudibert5793]

Brillant thank you!! To mappe these complex functions, how did you add the dimension of time to make a video?

[@vector8310]

Your composite functions were truly dazzling. I mean, domain coloring of tan(z^z)^ln(z) is really quite original and appealing, in my book at least. Inspiring work. The artistic qualities of complex functions really pop in you videos. First rate.

[@lemmaxiom]

Thank you!

[@hmdz150]

Amazing! Thanks so much for this beautiful video

[@kris-the-pee-cat]

It's too beautiful !

[@Number_Cruncher]

Thank you for the blender add on. This is very useful.

[@lemmaxiom]

No problem!

[@paulostipanov7682]

Will you release any of the songs in this video, they're a delight to listen to.

[@frankaz8937]

Marveleos presentation! . Best explanation and video to me about it. Thanks a lot!

[@madhuriwalinjkar3329]

Great explanation 👌

[@batoulmhawesh2863]

Dude !! Your video is so underrated !!

[@lemmaxiom]

Thank you - that means a lot to me!

[@TheNorrow]

Great job! and amazing soundtracks

[@AJ-et3vf]

Awesome video! Thank you!

[@파트라슈-o4l]

Great works. Rimemann must see your youTube

[@totanghosh7158]

Wow excellent.. I want to learn how did you do this animations? Which software you are using

[@surenmoodley7744]

Really good explanation. Thanks.

[@82Muntasir]

Great job 👏 Thank you 🙏

[@IgorDavydov91]

Such an amazing video! Thank you, mate!

[@Applied_Mathemagics]

Excellent!!!

[@MusicEngineeer]

math for artists!

[@beatboxanimationnandan]

Very very helpful

[@andreizonga4611]

Actually, for domain coloring, there is a way to visualise the modulus of z: either brightness or saturation (though brightness is mostly used, because that way, the zeros of the function are black and the infinities are white). Also, I would like to mention a couple of helpful and popular ways to visualise complex functions that you didn't mention: vector fields and Riemann spheres (which are basically just those animations at the beginning on a sphere). But apart from this, the video was great!

[@lemmaxiom]

Glad you liked it! Those are definitely some good visualization techniques worth mentioning.

[@stevewhitt9109]

to bad you were not around to give Jesus "good" advice.

[@peacemaker42069]

love the video man great work

[@prashanthkumar0]

I am high school student and want to request you something as this channel's motto is «Not only Math» Once i saw that the complex vector field of complex functions 1/z² represents an electric dipole which was shocking . and 1/z represents a monopole. My question is how could we use that thing to solve questions in physics and how do i find and scale the dipole moment of the field . it would be really pleasent to work with complex number since too easily we get field for charge and we can just shift and add functions to add new charges to the system and study about its field and potential ...

[@lemmaxiom]

Hi Prashanth, I thought it was interesting as well when I first saw that complex vector fields of certain functions resemble electric fields around point charges. As far as I am aware, though, there isn't much to be gained from trying to represent and manipulate electric / magnetic fields with single variable complex functions. In fact, one would not really need to use any single variable complex analysis for classical electrodynamics problems. But complex numbers in general obviously do have applications in physics (think Hilbert spaces and Schrödinger's equation in QM). In some areas of modern mathematical physics, multivariate complex analysis proves to be quite useful as well (in String Theory, for example).

[@prashanthkumar0]

@@lemmaxiom oh..ok I will definitely look at those stuffs after the exams :) ... complex numbers are really fascinating ...i love them just because how easy it is to understand yet being soo powerful ... The most interesting thing is Euler's representation.. I really want to explore it further and look at quaternions too as i like computer graphics :) Thank you so much ! I still donot understand quaternion but 2D rotations are simpler so expect 4D rotation to be the same xD I would be thankful if you made something on explaining quaternion . I watched 3b1b video 1-2 years back on the topic but didn't get it well :( I think i get more what you explain.

[@MrLegarcia]

Thank you

[@SimchaWaldman]

08:03 My favorite plotting. 11:40 Second in line.

[@purim_sakamoto]

とてもいいね

[@curtpiazza1688]

Cool stuff! 😂 Thanx!

[@WhiteFox-ce8ep]

What music did you use in the video? Is so relaxing

[@lemmaxiom]

Glad you liked it! I made each track myself using LMMS ( lmms.io/download#windows ), a free and open source DAW.

[@WhiteFox-ce8ep]

@@lemmaxiom I see, you are very talented

[@code.with.chirag]

thanks a lot for this video!!! is there any way instead of domain coloring we can add an image as an input and the function spits distorted image as an output?

[@lemmaxiom]

Definitely - you could let each pixel in an image be a point in the plane, and pass the entire image through a function. Every input pixel could get mapped to some other position (some other complex number) and the pixel at the output position could take on the color of whatever the input pixel was. Could be a good little coding experiment.

[@leonardolopez1511]

Where can i download the song that plays at 00:00 ?

[@rayannzakhia9372]

I really want to know the music that is used. Was hoping someone would know in the comments

[@lemmaxiom]

I actually made all the music myself! People seem to really like the tracks, I should probably upload them on their own when I get the chance. I have a link to the software I used in the description if you're interested.

[@rayannzakhia9372]

@@lemmaxiom definitely should upload them their very good

[@jeppejabob]

I love this! Keep it up :)

[@scienceuser4014]

i tried to install the blender, but i cant see or find the little box at 21:29 , is there some other place where i can find the function editor. thanks

[@scienceuser4014]

nevermind, it works now... Thanks for the video!

[@scienceuser4014]

actually, i found that the option menu dissapears if i click off the curve - and it wont come back unless i press add object - is there a setting i can enable that allows me to automatically bring this box up when i click on the curve?

[@scienceuser4014]

nevermind, it was f9 to bring it up.

[@yogeshbali7289]

Bro I want to do research in complex analysis I have studied some of the books Can you suggest me some books for going into research for complex

[@lemmaxiom]

I've read _Visual Complex Analysis_ by Tristan Needham - that could be a good spot to start. For something much more comprehensive and rigorous you could check out _Complex Analysis_ by Elias Stein and Rami Shakarchi. If videos help you, I highly recommend checking out Richard Borcherds' channel on KZbin (here's his complex analysis playlist: kzbin.info/www/bejne/p4m6g39sg62fjcU ) - he's a professional mathematician and has series of short lectures for a ton of topics.

[@andreizonga4611]

I don't know any books, but Mathemaniac is currently making a video series about complex analysis.

[@milessaye483]

quelle programmme avez-vous utilisé

[@lemmaxiom]

I used _Blender_ ( www.blender.org/ ) as well as the Python library _Matplotlib_ ( matplotlib.org/ ) for the plots. To edit the video I used Adobe Premiere Pro.

[@InquilineKea]

This is so psychedelic

[@wugionyoutube]

Great graphics, congratulations and thanks. A pity though that these powerful tools are not used for *real 4D* rendering, which would show the "true" geometry of the function (Ok, projected, but so are 3D renderings). My simple Graphing Calculator 4.0 *does offer full 4D*, allowing me to feed my own channel for Complex Functions: kzbin.info/aero/PL5xDSSE1qfb6FIk0Pl3VCg5p3Ema52hEG and other 4D beasts: kzbin.info/aero/PL5xDSSE1qfb6c7UHcURl6wXh0pH4ARB75

[@diribigal]

A little nitpick on the intro stuff, your formula arctan(Im(z)/Re(z)) is only going to work in quadrants 1 and 4 (and not on the y-axis obviously). In general you need something like en.wikipedia.org/wiki/Atan2 But nice video! It's more watchable than the other "plots of complex functions" videos I've seen floating around.