00:00 Intro 00:34 The [geometric] intuition for complex derivative 4:11 Producing the formal definition 5:19 Example 1. A linear polynomial in C 7:34 Example 2. A conjugate function
Just discovered your videos now as I have to refresh my memory on this topic. Wish this existed during my study! Super clear explanation, thanks for sharing this for free!
Complex derivative and examples? More like "Completely dynamite and expansive!" Thanks for helping to shatter our old ways of thinking and broadening our horizons.
Cool! BTW, another way to show how crazy this limit in the second example gets is to put the limit in the polar format using our old boy Euler's formula. The function that measures distance in the metric space returns a real number, and this is good because we want ordering. However, the price to pay, and the phantom that will haunt us forever on this, is that the phase(argument) information collapses and is lost, and strange things the average normal human mind has trouble accept at first start to happen.
I have a question about the linear approximation for f(z). I agree that the Delta function can (or does) exist, but it doesn't look right that it is equal to the differential limit. The limit imposes an infinitesimal difference between z and z0, while the linear approximation accepts any z, for a given z0. So my question is: where am I getting it wrong?
wouldn t the last example give the same results in a function that mirrors dots in cartesian plane because none of the process included some unique complex property or anything?? i rlly need an answer
That's the whole thing for "linear approximation". Delta is not hiding anything. It's just the non-linear part you would omit for the linear approximation.
I guess there is a typo in the representation of the linear approximation of complex functions. In the end of the formula, it is supposed to be z_0 instead of z?
if delta is defined so f(z)=f(z0)+(z-z0)*Delta(z), then why isn't Delta (z)= (f(z)-f(z0))/(z-z0) instead of the limit of that expression? I mean the derivative as the slope would still work if it is defined as the limit of delta and not delta itself. f(z)=f(z0)+(z-z0)*f'(z) is not correct in general.
I'm sorry but I simply can not accept the fact that taking the complex conjugate is not a differentiable map. I will need at least one week to wrap my head around that ridiculous sounding fact
It makes sense to me bc the complex conjugate is a very jarring thing. It reflects things about the real axis. That's not a very smooth operation. It causes the input to suddenly jump in another direction.
Please do the quiz to check if you have understood the topic in this video: tbsom.de/s/ca
00:00 Intro
00:34 The [geometric] intuition for complex derivative
4:11 Producing the formal definition
5:19 Example 1. A linear polynomial in C
7:34 Example 2. A conjugate function
Thanks :)
Just discovered your videos now as I have to refresh my memory on this topic. Wish this existed during my study! Super clear explanation, thanks for sharing this for free!
Glad it was helpful! :) It is for free but only because nice people support me :)
My brain is completely fried and I love it. Thank you for this series!
Complex derivative and examples? More like "Completely dynamite and expansive!" Thanks for helping to shatter our old ways of thinking and broadening our horizons.
Cool! BTW, another way to show how crazy this limit in the second example gets is to put the limit in the polar format using our old boy Euler's formula. The function that measures distance in the metric space returns a real number, and this is good because we want ordering. However, the price to pay, and the phantom that will haunt us forever on this, is that the phase(argument) information collapses and is lost, and strange things the average normal human mind has trouble accept at first start to happen.
Nice to see the next video so soon.
I really appreciate it.
Great video as always, excited for the series
3:54 shouldn't there be an error term for f(z)?
It's in Delta :)
Haa ! Je viens justement de finir mon thé ☕.
Bonne pause en perspective. 👌
I have a question about the linear approximation for f(z). I agree that the Delta function can (or does) exist, but it doesn't look right that it is equal to the differential limit. The limit imposes an infinitesimal difference between z and z0, while the linear approximation accepts any z, for a given z0. So my question is: where am I getting it wrong?
At z_0, it's equal to the differential limit :)
Incredible videos!!! Thank you so much!
Thanks for watching! :)
wouldn t the last example give the same results in a function that mirrors dots in cartesian plane because none of the process included some unique complex property or anything?? i rlly need an answer
Any function f: C → C is also also a function f: ℝ²→ℝ²
I love this playlist thanks for sharing
I'm glad you like it :) And thanks for the support!
Love it :) Thank you!
Awesome video! Thank you!
nice work 👍
really interesting, will you be covering modular forms?
Maybe at the end of the series :)
Why is this form linear aproximation before z is fixed to z0? Delta is just hiding nonlinearity until it is evaluated, this nomenclature confuses me.
That's the whole thing for "linear approximation". Delta is not hiding anything. It's just the non-linear part you would omit for the linear approximation.
I guess there is a typo in the representation of the linear approximation of complex functions. In the end of the formula, it is supposed to be z_0 instead of z?
I don't see a typo. Can you give a timestamp?
@@brightsideofmaths 3:19 delta(z) should be delta(z_o)?
@@sirlewis3975 No, it should be Delta(z).
SIR - WHETHER CONTOUR IS DOMAIN OF FUNCTION ? THANK U SIR
What?
Amazing 🤩🤩🤩
Thanks 🤗
We will really appreciate it 🙏 🙌
if you can teach us
Geometric Analysis
if delta is defined so f(z)=f(z0)+(z-z0)*Delta(z), then why isn't Delta (z)= (f(z)-f(z0))/(z-z0) instead of the limit of that expression? I mean the derivative as the slope would still work if it is defined as the limit of delta and not delta itself. f(z)=f(z0)+(z-z0)*f'(z) is not correct in general.
The limit comes in because Delta should be continuous at z_0.
9:27 is it f(z)=z!.
What is the question?
Complex factorial?
@@KM-om1hm Yeah, this could be defined, see Gamma function? :)
@@brightsideofmaths okay let's check it out, thank you
@@KM-om1hm Yeah, I have a nice video about it: tbsom.de/s/aoms
I'm sorry but I simply can not accept the fact that taking the complex conjugate is not a differentiable map. I will need at least one week to wrap my head around that ridiculous sounding fact
The next videos can help you: "complex differentiable" is just a very strong notion.
It makes sense to me bc the complex conjugate is a very jarring thing. It reflects things about the real axis. That's not a very smooth operation. It causes the input to suddenly jump in another direction.
This is BULLSHIT. No one has shown a complex number on a complex axis.
I even can put R^n on one axis :D
Great video as always, excited for the series