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)
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).
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
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?
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!
The integers or real numbers are self dual:- ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-AxPwhJTHxSg.html 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!
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!
The integers or real numbers are self dual:- ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-AxPwhJTHxSg.html 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!
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.
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.
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!
The integers or real numbers are self dual:- ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-AxPwhJTHxSg.html 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!
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)
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.
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!!!!!!
The integers or real numbers are self dual:- ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-AxPwhJTHxSg.html 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!
Great video! The style is nice, I love how you just.... walk and talk through things without aiming at the viewer being forced into a frame. . In Maths learning, I believe it is best that the student finds their own frame instead of always being provided it such as in primary school. As such, the student can modify their frame and still use the same lecture/video to find new things which are both true and Maths.
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)).
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.
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.
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!!
The integers or real numbers are self dual:- ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-AxPwhJTHxSg.html 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!
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.
The integers or real numbers are self dual:- ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-AxPwhJTHxSg.html 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!
The integers or real numbers are self dual:- ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-AxPwhJTHxSg.html 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!
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
The integers or real numbers are self dual:- ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-AxPwhJTHxSg.html 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!
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 :)
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?
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.
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.
aha! so the derivative of a complex function is the complex number you can multiply a small change of z with to get the actual transformation of the function so that the difference in the input z and dz is equal to the difference in the output z and dz
Not embarrassing at all!! That perspective of the real derivative is just never taught, but gives a great way to understand more intricate versions of the derivative :)
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)?
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.
The integers or real numbers are self dual:- ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-AxPwhJTHxSg.html 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!
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)
There is the easiest way to solve quadratics .....it surpasses even po-shen loh's method .....and this can lay foundation to solve cubic and quartic..... I can make a video on it if the response is good....
You said that scaling and rotating (-1+2i) according to (-2+4i) will give (-3-4i) but (-1+2i).(-2+4i)=(-6-8i) not (-3-4i) also Scaling (-1+2i) by 2 root 5 is 10 but magnitude of (-3-4i) is 5. I am not able to understand how the transformation leads (-1+2i) to (-3-4i)????
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
Thank you for your great videos And I want to ask you to: Please make bijection between: 1)irrational numbers and real numbers. 2)real numbers and complex numbers.
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?
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.
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
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
Nicely done, congratulations. With that "visually explained" though, I was given a wrong impression, expecting visuals of a surface representing a complex function, together with its tangent planes representing its derivative. These can be seen in the video *Derivatives* of my 4D complex functions channel, here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-zuwObgLTda8.html
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.
