What does it mean to take a complex derivative? (visually explained)

[@vcubingx]

Some extra info: At 20:47, I mention that a function is holomorphic if it satisfies the cauchy-riemann equations. There's an extra condition: the partial derivatives have to be continuous as well. For example, f(z) = {0 if z=0, z^5/|z^4| if z!=0} satisfies the cauchy-riemann equations but is not differentiable at z=0. Thanks to Ge for pointing this out! Mistakes: 3:15: The upper number line should still be labeled as "x" instead of "x^2" 14:56 [z^n]' = nz^(n-1)

[@mathmanindian]

Bro, improve your thumbnail Try making it attractive I really like your work !

[@josephkwong7388]

@@mathmanindian I kinda like the thumbnail tbh

[@probablyapproximatelyok8146]

Also, at 14:17, I think the example you gave for why the converse doesn’t hold seems off. If angles were preserved, then the arrows would stay at uniform angles from each other. But that clearly isn’t the case in the example you gave, since some arrows become close to 45° angles from each other, and others are clearly less than 30°. One example for why the converse doesn’t hold that is conformal but not complex differentiable at the origin is z -> conj(z).

[@vcubingx]

You're right! I can't believe I didn't catch that either. Basically, what I wanted to animate was a rotation matrix applied to the dz, and then a scaling factor applied to two opposite dz vectors. I think I instead scaled all of them :p, but I (hopefully) think it can somewhat get the point across

[@epicmorphism2240]

Another small mistake at 9:01 holomorph isn‘t equivalent to complex differentiable

[@mathemaniac]

Welp - you beat me to it! I was planning a video which will exactly be about CR equations, and is going to be the next video for my complex analysis series. Would you mind me linking this video in my own Essence of Complex Analysis playlist?

[@vcubingx]

I don't mind at all! Your videos are amazing, keep up the great work!!

[@mathemaniac]

​@@vcubingx Thanks! Will add that now.

[@glory6998]

I follow both of you

[@standowner6979]

@@glory6998 Okay.

[@tanchienhao]

You BOTH are awesome!! Competition/collaboration would do wonders for the youtube complex analysis videos landscape wonders :)))

[@alicesmith5361]

Wow, this is incredible! Now I understand way more about this than when I covered it in an independent project. Considering the differential, the condition of a linear mapping makes complete sense as you'd want any step away from the input to act in the same way (as it is being multiplied by a single number, the output of the derivative at that point) regardless of angle. What a wonderful video!

[@vcubingx]

Thanks! Glad it helped

[@hyperduality2838]

The integers or real numbers are self dual:- kzbin.info/www/bejne/d6mzqJuAia2rick Symmetric matrices (real eigenvalues) are dual to anti-symmetric matrices (complex eigenvalues) -- linear algebra, Gilbert Strang. Real numbers are dual to complex numbers. Complex numbers are dual. "Always two there are" -- Yoda. The spin statistics theorem:- Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- wave/particle or quantum duality. Bosons are dual to Fermions -- atomic duality. Duality creates reality!

[@aliscander92]

Brilliant! Great lecture! I'm radioelectronic engineer, so I regularly use complex functions theory in my calculations for radar applications. Your made me remember some details from our university course of complex functions. Thank you very much!

[@vcubingx]

Thanks! Glad you enjoyed it

[@hyperduality2838]

The integers or real numbers are self dual:- kzbin.info/www/bejne/d6mzqJuAia2rick Symmetric matrices (real eigenvalues) are dual to anti-symmetric matrices (complex eigenvalues) -- linear algebra, Gilbert Strang. Real numbers are dual to complex numbers. Complex numbers are dual. "Always two there are" -- Yoda. The spin statistics theorem:- Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- wave/particle or quantum duality. Bosons are dual to Fermions -- atomic duality. Duality creates reality!

[@unnamedemptiness2002]

Your pronuntiation has improved insanely bro, and you keep covering topics that nobody animated before, thanks for that

[@speeshers]

WOW, what an amazing intuition you developed for the ideas presented, and the visuals are top-tier. Thanks so much!!

[@Krageon-Offline]

I don’t know what it is about this music, but it for some reason gave me a sense of true calmness for once in the last ~3 years… keep it up.

[@frankansari3457]

This is really great stuff. From a real function you can always take a deriviative if the function has no gaps, jumps or poles. With complex functions you can not take it for granted that you can do this. This video explains why.

[@estebanmartinez4803]

Grwat video! Just to say that there is a little mistake at 14:56 The derivative of z^n obeys, as you say, the same rule as for real values, so it should be nz^(n-1)

[@vcubingx]

Oh god, how did I mess that up

[@Caleepo]

@@vcubingx I have a feeling you are gonna reupload this video :p. Anyways awesome vid tho.

[@ammyvl1]

@@vcubingx mixed it up with the integral lmao

[@NexusEight]

Fantastic visualisations! Some of the animations are rarely seen here on youtube, like the first most basic one, mapping the change in x to the change in y , each on their own number lines. Great work!

[@hyperduality2838]

The integers or real numbers are self dual:- kzbin.info/www/bejne/d6mzqJuAia2rick Symmetric matrices (real eigenvalues) are dual to anti-symmetric matrices (complex eigenvalues) -- linear algebra, Gilbert Strang. Real numbers are dual to complex numbers. Complex numbers are dual. "Always two there are" -- Yoda. The spin statistics theorem:- Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- wave/particle or quantum duality. Bosons are dual to Fermions -- atomic duality. Duality creates reality!

[@smorcrux426]

Oh my god! I knew literally nothing about this topic beforehand and I just thought about this question randomly yesterday, and now I feel like I understand this really well! Thanks a ton, you really do help people out.

[@Applied_Mathemagics]

This is the best video on KZbin on the subject. As good as (and please forgive me if the comparison if found insidious) 3Blue1Brown. KEEP IT UP!!!

[@bogdanmihai4599]

Mulțumim!

[@kamalalagarsamy2583]

I watched many videos, but they were not clear. This is the best explanation of complex functions.

[@willknipe2607]

BEAUTIFUL. really intuitive explanation for how the cauchy-riemann equations follow from a function being analytic. DIdn't really click until now!

[@SubAnima]

This is such a great video. My lecturer made it seem like the Cauchy-Riemann equations just fell from the sky, this gave me some beautiful intuition. Thank you!!!!!!

[@vcubingx]

Thanks!! Glad you enjoyed it

[@hyperduality2838]

The integers or real numbers are self dual:- kzbin.info/www/bejne/d6mzqJuAia2rick Symmetric matrices (real eigenvalues) are dual to anti-symmetric matrices (complex eigenvalues) -- linear algebra, Gilbert Strang. Real numbers are dual to complex numbers. Complex numbers are dual. "Always two there are" -- Yoda. The spin statistics theorem:- Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- wave/particle or quantum duality. Bosons are dual to Fermions -- atomic duality. Duality creates reality!

[@cheesecak11857]

Let's gooooo! Can't wait to watch this Vivek!

[@actualBIAS]

My friend. Thank you for visualizing this masterpiece. This just helped me to overcome the barrier i was stuck with.

[@tariqandrea398]

Me too.

[@shashwatbhatnagar659]

superb,I was searching the whole internet for this and you explained it in the most beautiful way possible

[@thatchessguy7072]

I’ve just started complex analysis this semester. This is very helpful.

[@brendawilliams8062]

I tried it for 20 years.

[@md.hamidulhaque5816]

What a video that was!!!!!! I completed my post-graduation in Physics from a third world county. Always wanted to get deeper intuition, and this video is just amazing. Be blessed always.

[@いむならむ]

Best ever!!! explanation on Cauchy Riemann equations of which "This matrix transformation can't be any linear transformation. It has to look like multiplying a complex number" has me convinced.

[@vcubingx]

Thank you!!

[@acamarocutcher8845]

Thank you for the effort you put into making these videos. It's helping appreciate complex analysis more.

[@tigranchtchyan1614]

Wow, a great video!! Brilliant ideas and illustrations! Thanks for your effort. P.S. I work with manim too, so I know how hard it is to make such animations.

[@vcubingx]

Thanks!

[@王劲飞-z4z]

Great video! Really inspred me when I am struggling to visually understand complex functions!

[@susanariveracabrera764]

Wonderful explanation and great video! Thank you so much for clarifying things to us. Keep on going with this great videos, they are awesome!

[@vcubingx]

Thanks! Glad you enjoyed it!

[@hyperduality2838]

The integers or real numbers are self dual:- kzbin.info/www/bejne/d6mzqJuAia2rick Symmetric matrices (real eigenvalues) are dual to anti-symmetric matrices (complex eigenvalues) -- linear algebra, Gilbert Strang. Real numbers are dual to complex numbers. Complex numbers are dual. "Always two there are" -- Yoda. The spin statistics theorem:- Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- wave/particle or quantum duality. Bosons are dual to Fermions -- atomic duality. Duality creates reality!

[@cmilkau]

when visualizing rotations, please consider breaking high rotational symmetries so the rotation angle is more obvious

[@lowerbound4803]

Your explanation is unreal!!! 💫💫

[@Spandan_Ghoshal]

Hats off to you 🙏🙏🙏 you have given immense amount of effort to make this video and i found this really really helpful... thanks again ❤️❤️❤️

[@darmok3171]

This is an awesome video! I've spent a long time trying to understand why certain "smooth looking functions" (not in the mathematical sense) are not complex differentiable. I was especially stumped by |sin(|z|)| * e^(i*arg(z)) and conj(z)·sin(z) + cos(conj(z)).

[@TheFallenTitan]

Lovely Video! Thank you so much, very well explained. I wish you will make a video on Wirtinger Derivatives--generalizing derivatives to non-holomorphic functions!!

[@hyperduality2838]

The integers or real numbers are self dual:- kzbin.info/www/bejne/d6mzqJuAia2rick Symmetric matrices (real eigenvalues) are dual to anti-symmetric matrices (complex eigenvalues) -- linear algebra, Gilbert Strang. Real numbers are dual to complex numbers. Complex numbers are dual. "Always two there are" -- Yoda. The spin statistics theorem:- Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- wave/particle or quantum duality. Bosons are dual to Fermions -- atomic duality. Duality creates reality!

[@jimlbeaver]

Excellent video…very clear and well-paced. Nice job and thanks!

[@tanchienhao]

your channel is awesome!!! keep the great videos coming! i would love to see some info on riemann surfaces and their classification if u are into that :)

[@sour5blue]

Woah i never learned the intuition for calculus in complex numbers

[@isaigordeev]

great job and keep going at the moment you decided to do this kind of stuff you definitely did not mess up :) also would like to see something advanced about conformal maps on the complex plane

[@hyperduality2838]

The integers or real numbers are self dual:- kzbin.info/www/bejne/d6mzqJuAia2rick Symmetric matrices (real eigenvalues) are dual to anti-symmetric matrices (complex eigenvalues) -- linear algebra, Gilbert Strang. Real numbers are dual to complex numbers. Complex numbers are dual. "Always two there are" -- Yoda. The spin statistics theorem:- Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- wave/particle or quantum duality. Bosons are dual to Fermions -- atomic duality. Duality creates reality!

[@mnada72]

Big thank you. This was really helpful specially that Cauchy-Reimann equation is a consequence of Jacobian matrix.

[@_tgwilson_]

Superb video. More please! It really helped me with some of the concepts in The Road to Reality. I'm sure Roger Penrose would love it : )

[@hyperduality2838]

The integers or real numbers are self dual:- kzbin.info/www/bejne/d6mzqJuAia2rick Symmetric matrices (real eigenvalues) are dual to anti-symmetric matrices (complex eigenvalues) -- linear algebra, Gilbert Strang. Real numbers are dual to complex numbers. Complex numbers are dual. "Always two there are" -- Yoda. The spin statistics theorem:- Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- wave/particle or quantum duality. Bosons are dual to Fermions -- atomic duality. Duality creates reality!

[@carmelpule8493]

When considering complex differentials, we could consider navigation and directions followed on a field. If one is following a path where each position is a vector then the differential is the present position ,minus the old position divided by the time taken ( the function is with respect to time. Hence the rate of change of the walk in this situation. If we consider a field where wheat is growing , each stub of wheat is the vector field and if we subtract two nearby stubs of wheat in their vector form we get the rate of change of the vector field of wheat, which has magnitude and direction. The important issue is to understand what is RATE OF CHANGE with respect to some variable.

[@mridulk81]

that explanation of the jacobian....ohhhhhhhhhhhhh my goood THANK YOU 😭😭

[@leotimm6805]

It's always a great pleasure to watch your videos ! Thank you so much !!!

[@vcubingx]

Thanks for watching!

[@isakhammer6558]

So good quality, you are great! I may also appreciate a video about laurent series and the relations to tayler expansions!

[@vcubingx]

Thanks! As for Laurent series, maybe, I'll have to see. I added it to my list of topics though!

[@davidhicks8290]

Underrated! Amazing video thank you!

[@nafrost2787]

So because complex differentiability requires that the linear transformation we use to approximate the function to consist solely from scaling and rotating, and because we can always convert a function from a domain of C to a domain of R^2 bijectively, can we say that complex differentiability is a stronger property of a function than regular differentiability? Which allows the linear transformation we use to approximate the function to be any linear transformation?

[@vcubingx]

Ignore my previous reply, from my understanding yes, complex differentiation is a stronger property than differentiation over R^2 -> R^2

[@dirichlettt]

Needham will always be my favorite complex analysis book

[@vcubingx]

It's amazing!

[@rikdebbanerjee55]

The complex idea of amplitwist!!! 😍😍

[@SamiulIslam-vv5vc]

It was really a great one!!! I really loved it!!!

[@Abbas-fl3bw]

the animations are so smooth bro wtf

[@AJ-et3vf]

Awesome video! Thank you!

[@rxphi5382]

Those are some beautiful animations!

[@ANSHUL-n7l]

19:57 can somebody explain why we put two matrix equal?

[@Neme112]

13:26 I don't get this. It seems like the angles are not preserved. For example, the angle between the x and y axes is initially 90 degrees, but it grows to 180 degrees.

[@valor36az]

Amazing explanation

[@bulat314]

Amazing! This helps a lot👍

[@BigEpsilon]

Very insightful. Thank you.

[@chingizarystanbekov151]

amazing work,

[@ycombinator765]

لیجنڈ واپس آگیا ہے. ❤️❤️🌹

[@Zonox-ml4jq]

¿Which software do you use? it's amazing, i mean, i'd love to try myself and dig into complex functions!

[@s.m.m99203]

Hi. Thank you. May I ask how you make such animations?

[@Djake3tooth]

this is so much more fun to watch when i need to do homework

[@RohanDasariMinho]

Great work!

[@suyashpatni4032]

brilliant video!

[@izzapz]

Great video! Do you use any particular software to graph the plots in the videos?

[@vcubingx]

Sorry for the late reply, but I use manim! Check the desc for the code

[@izzapz]

@@vcubingx thanks!!!

[@Akshaylive]

@14:45 has an error in the last equation

[@lagrangian143]

will you do videos on harmonic analysis and operator theory?

[@vcubingx]

Maybe! I was planning a video covering some topics from harmonic analysis, but it's a tricky one to make, so I may put it off

[@gauthierruberti8065]

Thank you so much

[@bomboid]

3b1b what happened to ur voice?

[@VisuallyExplained]

Hey there, nice video! For reasons I can't really explain, I really like the title. :-)

[@vaguebrownfox]

Heyy, did anyone tried to figure out the proof for equations in 14:49 ?

[@valerierit2003]

Nice.. Plss do post frequently

[@djredrover]

Wow, Grant's visualization software is all over youtube!

[@mauriciocaviedes4552]

Great, great video! I didn't get the deriv of e^z. Tomorrow I'll try again.

[@animewarrior7]

thanks a lot brotha!

[@isakdupreez6201]

If you consider a complex differentiable function as a 2D vector field over the same 2D domain, the real part of the derivative is divergence and the imaginary part of the derivative is curl (which in 2D can be defined as a signed scalar)* * Except that both are scaled by a factor of 2.

[@angeldude101]

If you try doing this with a 3D vector, what you end up with is a quaternion as the derivative, with the imaginary curl being the 3 "vector" components.

[@NovaWarrior77]

Vivek with the sponsorships!!!!

[@HarshaJK]

At @3:57 the top line should be of x and not x^2

[@peterecco]

apologies, duplicated

[@swastikkalsi9586]

3blue1brown and now this🤩

[@luphiax4239]

How did you come up with that!!! you are a genius

[@christophem6373]

do hope you could illustrate complex integration !!! Thank you a lot !

[@vcubingx]

That's the plan! I mainly want to cover the Cauchy Integral Theorem and the Residue Theorem, and how it can be used to evaluate improper integrals

[@蒋正-k6u]

very good video, approaching 3b1b level

[@tedsheridan8725]

Great video! Question I've always had: It seems if you take any real, differentiable differentiable function f(x), and make it complex, i.e f(z), you get a holomorphic function. Is this an 'if and only if' condition? In other words can every holomorphic function be thought of as f(z) for some real differentiable function f(x)?

[@dng88]

Is the upper line x not x^2?

[@bennicholl-kv4ex]

how do you choose u and V vector functions?

[@proxxyBean]

Is there a way to use animation to visualize the output space using the timeline to stand in for the imaginary part?

[@ronakpatel6530]

Damn dawg you explained the shit outta that topic good

[@gyanvarshnay8053]

Well explained, though may I ask why is the presentation style so similar to 3blue1brown? Is it a new channel you created? Or are you another person who has taken inspiration from him

[@vcubingx]

I use his animation library

[@gyanvarshnay8053]

@@vcubingx I see, cool video!

[@hannesstark5024]

Fantastic!

[@peterecco]

surely the top line from 03:15 is just x, not x squared

[@geoffrygifari3377]

One thing i'm a bit confused about with conformal map in this video is that its definition implies angles are preserved, but to preserve angles you need to have crossing lines to form those angles. complex function maps a set of points in the complex plane to another set of complex points. does conformality imply that we define (arbitrary) line equations first in the complex plane, then the function preserves the angles between those lines?

[@monny1815]

Essentially, the point is that, zooming in very close to a point, the function will look like a linear transformation, which sends lines into lines. Now take two arbitrary lines, as you said, and look at them close to their point of intersection ,these will form an angle between their direction vectors. The fact that the linear transformation rotates every vector at the same rate, implies that it rotates the line vectors at the same rate hence the angles are preserved. Note that this is a local property and not global, in general a complex derivative will NOT send lines into lines, but zooming close enough this will happen, and if we look at the portions of lines then the angles of those portions of lines will be preserved.

[@vcubingx]

Right basically what Monny said. If a function is conformal at a point, the zoomed in transformation preserves angles as well - this means that for any choice of curves intersecting at that point, the angle (here, angle is the tangent angle) is preserved

[@vanadium4603]

what is the song in the background?

[@vcubingx]

It's in the description

[@pawejedrejko7398]

What is the function (of time) you use to represent the dynamics of e^z mapping?

[@sitrakaforler8696]

great job m8

[@phoenix2464]

7:50 has been scaled by roughly 1.15 ... should correct that

[@tuongnguyen9391]

Is this wirtinger calculus ?

[@prometheus7387]

A short way to summarize it: It's complex

[@ゾカリクゾ]

Top quality.

[@rudranshgoel3301]

Aren't cauchy reimann equations just necessary condition and not sufficient for a function to be complex differentiable. (This is what my prof told in the course on complex analysis)

[@Ganerrr]

anyone else kinda hoping he would try and somehow explain the C'th derivitive of a function, sorta like how you can take the 0.5th derivitive lol

[@connorhayes2374]

he has

[@vcubingx]

I've covered this already! Check out my "fractional derivative" video from a couple years back. Although I doubt I'll cover topics like that again, it's ridiculously hard to come up with good visual intuition for those topics

[@balasavenedintulashabalbeoriwe]

Hey, can you please help me? I am with you until 13:37 I'm not able to see how the angles involving the origin are preserved (it seems like pi/2 angle becomes pi) is this because the derivative is 0 there or some other reason? Thank you

[@balasavenedintulashabalbeoriwe]

Oops, you answered my question I just had to watch until 14:05 lol

[@vcubingx]

Nice! Glad it clicked in :)

[@manstuckinabox3679]

I always felt complex derivatives were highly similar to the divergence of a vector feild.

[@GoogleUser-ee8ro]

is holomorphic the same as analytical?

[@GoogleUser-ee8ro]

this video is such an excellent explanation of complex differentiation and Cauchy-Riemann equations that every engineering student or high school kid should watch it. In 25 min (or more if you watch it repeatedly) you will understand the mathematical intuition behind the beautiful visualization.

[@agrajyadav2951]

Bruh ur a legend