Your channel is really helpful. I am using it now to learn complex variables in 5th sem and i also watched a lot of your videos about calculus of variations during my 3rd semester. Just wanted to say thank you.❤
since f(z)=y+xi and dz=dx+dyi then when we write it as f(z+dz) shouldn't the real part add to the real part and the imaginary part add to the imaginary part to form f(z+dz)=(y+dx)+(x+dy)i Please explain this part will be very Thankful!
I thought the same, but just imagine it as a small nudge or a push in the function, since it’s two dimensional , it is in both directions, nothing to do with a + delta a because it is inside a function, all we are saying is a slight change in the function f(z) ie f(z+ delta z), so z = 2y + xi; small change i.e 2(y+deltay) and x+ delta x, delta i.e. small change
Great lecture! But I'd like to state, at 7:41 it seems to me there's a slip of the tongue, you state z=x+yi, which is the general case yes, but our defined z is in the form y+xi instead, then you go on saying delta(z)= dx + dyi, but you clearly write delta(z) as dy+dxi; if what I'm saying is untrue, I might be missing a crucial point and I'd very much appreciate a clarification. Cheers!
The y + xi you mention is actually f(z) (the 2y + xi), which is a function of z and not z itself. As a result, delta f is 2*dy + dx*i not dx + dy*i. The function f(z) can be anything, but z is always x + y*i. Hope that helps!
Oh right, thanks for the reply! So it essentially goes like this? z=x+iy as always, f(z)= a*z where ''a'' is a complex scalar, namely a= ( 2y+ix / x + iy ) ).
Kind of. Keep in mind that f(z) is not explicitly a function of z, because your 'scalar' a isn't really a constant; it depends on x and y. Since f(z) is not explicitly a function of z, we can't write it exclusively in terms of z without including x and y (which z depends on) in addition. As a result, f(z) = 2y+xi isn't complex-differentiable, so saying that 'f(z)= a*z where ''a'' is a complex scalar, namely a= ( 2y+ix / x + iy ) )' isn't technically wrong, but it's a notation I would caution against since you don't want to get the idea that f(z) = 2y+xi is *exclusively* a function of z.
At 7:40, why is """z.delta = x.delta + y.delta*i""" if, in the above function """f(z) = 2y + x*i""", the imaginary part (the part tied to the i) is """x""". Why is it not """z.delta = y.delta + x.delta*i""" instead? Please note that you still get to a similar answer either way (that the function is not holomorphic) but just want to make sure it's as accurate as possible! (Also, your videos are the bomb!)
z represents the standard complex variable, and is ALWAYS x + y*i (it follows that delta z is delta x + i* delta y as stated in the video). Just because my function is weirdly arranged with the 2y in the real part and x in the imaginary part doesn't mean that z (the standard complex variable) will suddenly have its definition changed. You're right that delta f would be 2*dy + dx*i not dx + dy*i. However, z is always x + y*i. Hope that helps!
for question2 please explain the expansion step for f(z+🔺z) the simplification is not making sense to me..... like why is 🔺x now associated with the complex i and why is there a coefficient of 2 for 🔺y ??
about f(z + h ), z is a complex but h is not , dont you need to do z cube with delta z too? h seems only the real part of z i am not a math majoring person. i just looked up holomorphic it said it means entire form. in math, complex differentiable. what word do you have for a singleform in math, what does it mean?
9:12 why do they have to equall? i don't see the motivation behind this restriction. You explained, that f(z+h) can approach f(z) from all sides, but so can f(x+dx, y+dy) in real analysis. and we have no problem with f_x=/=f_y
Well, f(z) is a function of z (a complex number). A complex number is like a 2 dimensional real number, since z = x + yi, so if I'm differentiating with respect to a complex number, I need to make sure that the limit as h -> 0 of delta f/delta z exists and is equal no matter which direction h approaches zero on a plane. For the f_x and f_y that you mention, you're looking at the derivatives with respect to one-dimensional numbers (i.e. x and y alone exist on a number line, but z exists on a plane). As a result, you are allowed to have f_x=/=f_y since the partial derivatives with respect to x/y are irrelevant to each other; you only need the partial derivative to exist from 2 directions for 1-D numbers like x and y. df/dz, however, is a derivative with respect to a 2 dimensional number, so you need the condition in 9:12. Hope that helps, and if you have any more questions, please ask!
If I would be inventing complex analysis i would've never thought of this requirement df/dz=(a definite function that doesn't depend on the type of dz). I would rather treat f as a multivariable real function with a vector output and df/dz as a directional derivative. But they might have a reason for it. Thank you for the reply)
It's still possible to use your convention to describe complex functions (to a limited extent), but in that case, we'll still have to keep in mind that the derivative is with respect to a 2-dimensional number/'vector' z, instead of the simple partial derivatives with respect to the scalar components x/y. Nevertheless, I'm glad I could help!
z = x +iy in cartesian form. so in the graph , we take y axis as imaginary part, which is perpendicular to the base. Is it not possible for both of the axis to be imaginary? And why dont we take the base as imaginary?
How did you add f( z + delta z) and get that value because if you go the way you are supposed to go to giving the equation for delta z then your answer is wrong. I can only get the answer you got if i make x the imaginary part and y the real part
I've explained this above in another reply but there's a difference between my function of z (which is 2y + x*i) and z itself, which represents z = x + y*i. Therefore, delta z = delta x + delta y * i. When you say 'make x the imaginary part and y the real part', what you end up with is delta f, NOT delta z. Hope that clears it up!
At 5:00, I explain that a holomorphic function should be differentiable from all directions (up, down, left, right, all diagonal directions, which would imply 360 degrees). The 8 directions I drew in the figure were just there to illustrate the idea that the function has to be differentiable from all directions.
@@FacultyofKhan Thank you for the response. I'm going to ask one last question to clear all doubts: are you saying that being differentiable from up, down, left, right, and diagonal directions suffices to say the function is differentiable from all directions (if so why?) OR, are you saying "up, down, left, right, and diagonals" is just a shortcut to saying all 360 degrees?
The latter: I obviously can't mention all 360 degree directions in a short sentence. So "up, down, left, right, and diagonals" is just a shortcut to saying all 360 degrees. Being differentiable from 8 directions doesn't necessarily mean differentiable from all directions.
A combination of Churchill and Mathematical Physics by Boas - there's other sources I use here and there to supplement my explanations but those two are the core ones for this playlist.
![Imaginary Numbers Are Real [Part 1: Introduction]](http://i.ytimg.com/vi/T647CGsuOVU/mqdefault.jpg)
Man you are so good at explaining. You have a super power to transmite knowledge to our minds!! Thanks for the awesome playlist on Complex Functions.
Thank you so much!
El arvpez It’s a great substitute for 3blue1brown for the topics he (3b1b) doesn’t discuss!
This is the one of the best remote courses I HAVE EVER had!
thank you very much for this series. our professor goes very fast so I have a hard time keeping up. These videos really helped me!
How did the rest of your class go?
What a channel. I feel like I found a gold nugget weeeee :3
It's too late though, but tomorrow I swear I'll binge
these sorts of video lectures are what save me whenever I fall back in my course. Thanks for your great content :D
Can't believe I just found this! Thank you sooo much! :D
Glad to help!
Your channel is really helpful. I am using it now to learn complex variables in 5th sem and i also watched a lot of your videos about calculus of variations during my 3rd semester. Just wanted to say thank you.❤
How have your studies been going?
I love this format of lectures. I hope you make more videos on this topic.
I'm a senior (high school) and you make this easy to understand. Very good channel.
Thank you!
Wait you see these things in high?
Yup, it pops as part of physics
Nicely explained mate, I'm lucky that I've found your channel
Thank you so much. You are my heros for my AMATH501 class!
Complex functions? More like “Come on, these are some fun expositions!” Thanks again for making and sharing this amazing course.
Glad you enjoyed it!
I am an undergrad who is about to take a complex analysis course. This is great!!
Thank you for the series of lectures.
Glad you like them!
Great class! You were very didactic and organized! Plus, you have a pretty handwriting. Thanks !
Very Nice. Keep up the good work. I 100% disagree that Mathematics is a dry subject. Mathematics is beautiful.
since f(z)=y+xi and dz=dx+dyi then when we write it as f(z+dz) shouldn't the real part add to the real part and the imaginary part add to the imaginary part to form f(z+dz)=(y+dx)+(x+dy)i Please explain this part will be very Thankful!
I thought the same, but just imagine it as a small nudge or a push in the function, since it’s two dimensional , it is in both directions, nothing to do with a + delta a because it is inside a function, all we are saying is a slight change in the function f(z) ie f(z+ delta z), so z = 2y + xi; small change i.e 2(y+deltay) and x+ delta x, delta i.e. small change
Great! Learned a lot in 10 min! 😂
Wow wow! This was so fabulous
math is fun due to professor like you.
Great lecture! But I'd like to state, at 7:41 it seems to me there's a slip of the tongue, you state z=x+yi, which is the general case yes, but our defined z is in the form y+xi instead, then you go on saying delta(z)= dx + dyi, but you clearly write delta(z) as dy+dxi; if what I'm saying is untrue, I might be missing a crucial point and I'd very much appreciate a clarification. Cheers!
The y + xi you mention is actually f(z) (the 2y + xi), which is a function of z and not z itself. As a result, delta f is 2*dy + dx*i not dx + dy*i. The function f(z) can be anything, but z is always x + y*i. Hope that helps!
Oh right, thanks for the reply! So it essentially goes like this? z=x+iy as always, f(z)= a*z where ''a'' is a complex scalar, namely a= ( 2y+ix / x + iy ) ).
Kind of. Keep in mind that f(z) is not explicitly a function of z, because your 'scalar' a isn't really a constant; it depends on x and y. Since f(z) is not explicitly a function of z, we can't write it exclusively in terms of z without including x and y (which z depends on) in addition. As a result, f(z) = 2y+xi isn't complex-differentiable, so saying that 'f(z)= a*z where ''a'' is a complex scalar, namely a= ( 2y+ix / x + iy ) )' isn't technically wrong, but it's a notation I would caution against since you don't want to get the idea that f(z) = 2y+xi is *exclusively* a function of z.
Roger! That really helped thanks! As a 2nd year Physics undergrad. I'm looking forward to my free times to check your QM Math Lectures!
@@FacultyofKhan is it delta f or delta z
At 7:40, why is """z.delta = x.delta + y.delta*i""" if, in the above function """f(z) = 2y + x*i""", the imaginary part (the part tied to the i) is """x""". Why is it not """z.delta = y.delta + x.delta*i""" instead?
Please note that you still get to a similar answer either way (that the function is not holomorphic) but just want to make sure it's as accurate as possible! (Also, your videos are the bomb!)
z represents the standard complex variable, and is ALWAYS x + y*i (it follows that delta z is delta x + i* delta y as stated in the video). Just because my function is weirdly arranged with the 2y in the real part and x in the imaginary part doesn't mean that z (the standard complex variable) will suddenly have its definition changed. You're right that delta f would be 2*dy + dx*i not dx + dy*i. However, z is always x + y*i. Hope that helps!
very interesting video on knowledge transimission.Very wonderful I appreciate it
Awesome Sir, very helpful material and easy way to disperse knowledge
I know I'm late but there is a mistake at 7:55.
You took delta x under factor of 2 but it is just delta x
Huh? I don't see the mistake - it's just delta x in the expression for f(z+delta z).
Sorry my bad
Brilliant !
for question2 please explain the expansion step for f(z+🔺z) the simplification is not making sense to me..... like why is 🔺x now associated with the complex i and why is there a coefficient of 2 for 🔺y ??
You should have been my professor for this course
about f(z + h ), z is a complex but h is not , dont you need to do z cube with delta z too? h seems only the real part of z
i am not a math majoring person. i just looked up holomorphic it said it means entire form. in math, complex differentiable. what word do you have for a singleform in math, what does it mean?
Well, you don't have to evaluate the real and complex parts of h^3 (or h^2, for that matter) since you're making h approach zero, anyway.
great video. Thanks!
9:12 why do they have to equall? i don't see the motivation behind this restriction. You explained, that f(z+h) can approach f(z) from all sides, but so can f(x+dx, y+dy) in real analysis. and we have no problem with f_x=/=f_y
Well, f(z) is a function of z (a complex number). A complex number is like a 2 dimensional real number, since z = x + yi, so if I'm differentiating with respect to a complex number, I need to make sure that the limit as h -> 0 of delta f/delta z exists and is equal no matter which direction h approaches zero on a plane.
For the f_x and f_y that you mention, you're looking at the derivatives with respect to one-dimensional numbers (i.e. x and y alone exist on a number line, but z exists on a plane). As a result, you are allowed to have f_x=/=f_y since the partial derivatives with respect to x/y are irrelevant to each other; you only need the partial derivative to exist from 2 directions for 1-D numbers like x and y. df/dz, however, is a derivative with respect to a 2 dimensional number, so you need the condition in 9:12.
Hope that helps, and if you have any more questions, please ask!
If I would be inventing complex analysis i would've never thought of this requirement df/dz=(a definite function that doesn't depend on the type of dz). I would rather treat f as a multivariable real function with a vector output and df/dz as a directional derivative. But they might have a reason for it. Thank you for the reply)
It's still possible to use your convention to describe complex functions (to a limited extent), but in that case, we'll still have to keep in mind that the derivative is with respect to a 2-dimensional number/'vector' z, instead of the simple partial derivatives with respect to the scalar components x/y. Nevertheless, I'm glad I could help!
Definitely great!
the lectures are good
Marvelous💯💯
hope you were my professor. fascinating
Very impressive
Also, condition for a function to be holomphic is that df/dz* = 0. This is can easily be proven 🙂.
z = x +iy in cartesian form. so in the graph , we take y axis as imaginary part, which is perpendicular to the base. Is it not possible for both of the axis to be imaginary? And why dont we take the base as imaginary?
Thank you
How did you add f( z + delta z) and get that value because if you go the way you are supposed to go to giving the equation for delta z then your answer is wrong. I can only get the answer you got if i make x the imaginary part and y the real part
I've explained this above in another reply but there's a difference between my function of z (which is 2y + x*i) and z itself, which represents z = x + y*i. Therefore, delta z = delta x + delta y * i. When you say 'make x the imaginary part and y the real part', what you end up with is delta f, NOT delta z. Hope that clears it up!
I am looking for your Advanced Complex Variables playlist but I don’t see it. Can you help me?
Here you go! ua-cam.com/play/PLdgVBOaXkb9AzBcO4b-iRtGzSF0Km8ZoT.html
Hi,are x &y both independent variables? Or is y a function of x?
They're independent. 'y' is the imaginary part and 'x' is the real part of z.
Could you add a subsection of deformation + Cauchy Goursat
❤
Why is holomorphism defined by having these 8 directional derivatives? Why not 360 degrees?
At 5:00, I explain that a holomorphic function should be differentiable from all directions (up, down, left, right, all diagonal directions, which would imply 360 degrees). The 8 directions I drew in the figure were just there to illustrate the idea that the function has to be differentiable from all directions.
@@FacultyofKhan Thank you for the response.
I'm going to ask one last question to clear all doubts: are you saying that being differentiable from up, down, left, right, and diagonal directions suffices to say the function is differentiable from all directions (if so why?) OR, are you saying "up, down, left, right, and diagonals" is just a shortcut to saying all 360 degrees?
The latter: I obviously can't mention all 360 degree directions in a short sentence. So "up, down, left, right, and diagonals" is just a shortcut to saying all 360 degrees. Being differentiable from 8 directions doesn't necessarily mean differentiable from all directions.
Does someone know in what book these collection of videos in complex functions is based on?. Thanks
A combination of Churchill and Mathematical Physics by Boas - there's other sources I use here and there to supplement my explanations but those two are the core ones for this playlist.
Thanks
Thank you.
Which software do you use to write on screen?
Wonderful lecture!
Im at second year of applied mathematics in Ukraine, and I need to watch lectures in foreign language (english) to understand my subject.
Same, only physics)
redy5
That must be twice as difficult.
Got B on 'Complex variable functions theory' (the subject I needed to watch those videos for)
well, you are learning twice as much in the same time ;)
What bout functions of many complex variables
Where are you from, and are you a teacher ?
I'm in Canada and I used to be a teacher while in graduate school but not right now.
👍👍👍👍
Hello Dr how can i write like you :) i want to teach some ppl programing :)
What are u and v
@Supian Mat Salleh
but it would be nice if you write a bit slowly
Thank you! I'll keep that advice in mind for future videos!
Makethe video speed slower? :)
fajne
cant hear sheesh
the audio quality is awful
Sorry lol; I didn't have very good mics back then (I used my old laptop's mic to record)