I tracked down the origin of the word 'holomorphic'. It turns out that Briot/Bouquet (two students of Cauchy) introduced it in the 1875 second edition of their book about elliptic functions. The previous edition of their book had adopted Cauchy's term synectic, but in the second edition they wanted to focus more on meromorphic functions (which didn't previously have a name). Their stated explanation for the names is that a holomorphic function "resembles an entire function" (whole) in a particular region of the complex plane, while a meromorphic function "resembles a rational fraction [of entire functions]" (part) in a particular region of the complex plane. That is, the "-morphic" part is intended to mean "resembles" or "is shaped like", so that "holomorphic function" should read as "whole-type function" or "resembling a whole function", rather than "whole form". Your confusion about this inspired me to add the relevant clarification to Wikipedia, whence it will hopefully percolate into modern sources. (The original explanation is repeated in multiple 19th century sources I found, whose authors directly looked at Briot/Bouquet's book, but seems to have been lost in recent sources.)
when i was first learning this a few months back, the cauchy riemann equations were taught completely differently. However when you started off with the definition of real differentiability being whether or not you are able to approximate the function linearly made them feel so natural, like they just pop out, it's so nice
It's fascinating that the only reason why complex functions are infinitely differentiable if they're differentiable once, unlike differentiable functions from R^2 to R^2, is due entirely to complex multiplication. That's it. It gives us the CR-equations, and then everything else follows.
That's half the reason. The other half is that the functions are defined over a plane, and you can't have nondifferentiability when you are surrounded by complex differentiability unless you blow up. To see that this is essential, take any real valued function f(x) which is continuously differentiable once, but nowhere differentiable twice (integrate a nowhere differentiable function), and extend it linearly away from the real axis by defining f(x+iy) = f(x) + i f'(x) y. This function obviously isn't going to be holomorphic on any open set, but it is once complex differentiable on the real axis.
The best thing about Sir Richards is he knows these things in very depth and for him these topics are simple but he still teaches with such enthusiasm and excitement.
wow, when I learned the notation df/dx never bothered to question it, and because of that it seemed some properties were just thrown there, but with a lim makes so much sense, thanks for sharing so much :)
Maybe it is called holomorphic - "holo" -> entire and "morph" -> shape i.e the shape remains entire after the transformation by such functions - which is kind of true as these functions locally preserve angles when their derivatives does not vanish.
In general, satisfying the Cauchy-Riemann equations is not sufficient for complex differentiability. If the derivatives of the real and imaginary parts are continuous, however, then it should be the case.
Thank you so much for this fabulous lecture! And also, if possible, I would very much like to see "topological or integration free" proof of the the fact that "derivative of holomorphic fuction is continous". Ahlfors said at the begining of chapter 4 in his book that such a proof has been found, but doen't present it in the book. And I also don't see such a proof in other complex analysis textbook. Thank you so much again!
I think the examples at the very end of the video are not correct. For instance, x and y are certainly not holomorphic as they are Re(z) and Im(z). Some examples there could be z or z^2 = x^2-y^2+2xyi, ... Am I right? or I'm missing something??
I think what is meant is either of the two functions: f(z) = Re(z), or the function, f(z) = Im (z). Input ANY complex variable and the function truncates either the real part or the imaginary part and outputs what remains. This type of truncating function would not be holomorphic.
Here he's specifying a single function, f, from the complex plane into the real numbers, essentially defining a function from the complex numbers into the real axis. He's using "holomorphic" as "which polynomials can be considered as real parts of a holomorphic function?"
Great lecture, is a pleasure to hear a Professor talking about Complex Analysis and address the fact that a Holomorphic function is not exactly equal to Analytic function. Great UA-cam channel.
This is my favorite math channel to listen to in headphones while at work. Richard Brocherds is a great expositor, and unlike a lot of lecturers does a great job speaking mathematics, and making the ideas on the board audible. It is difficult (at least for me) to learn mathematics "by ear", but I am very grateful for these videos as impressively clear and followable examples of mathematics teaching in the medium of audio lectures (a technique introduced by Vivienne Malone Mayes, one of the first Black woman math PHDs).
3.10 is an epiphany to me. Never did I get it, but learned to use it. In the case of a real variable I still dont understand how it makes sense to split up the symbol df/dx when using, say, 'integration by parts'. Would the limit part follow the bottom half of df/dx around for it to make sense?
As Richard points out, this is not actually what it means and is just shorthand notation for the partial derivatives with respect to x and y which he wrote down. Writing d/d(z bar) and z in terms of x and y you will get the result. It's slightly dangerous notation because it can lead you astray, but once you get used to it it can be a handy way of seeing at a glance whether a function is holomorphic or not.
3:02 I think the standard notation for difference is \Delta. The notation d itself means "limit of delta"... Basically all textbooks of physics I know use such convention...
The word "holomorphic" was introduced by two of Cauchy's students, Briot (1817-1882) and Bouquet (1819-1895), and derives from the Greek ὅλος (holos) meaning "entire", and μορφή (morphē) meaning "form" or "appearance".[10]
I tracked down the origin of the word 'holomorphic'. It turns out that Briot/Bouquet (two students of Cauchy) introduced it in the 1875 second edition of their book about elliptic functions. The previous edition of their book had adopted Cauchy's term synectic, but in the second edition they wanted to focus more on meromorphic functions (which didn't previously have a name). Their stated explanation for the names is that a holomorphic function "resembles an entire function" (whole) in a particular region of the complex plane, while a meromorphic function "resembles a rational fraction [of entire functions]" (part) in a particular region of the complex plane. That is, the "-morphic" part is intended to mean "resembles" or "is shaped like", so that "holomorphic function" should read as "whole-type function" or "resembling a whole function", rather than "whole form". Your confusion about this inspired me to add the relevant clarification to Wikipedia, whence it will hopefully percolate into modern sources. (The original explanation is repeated in multiple 19th century sources I found, whose authors directly looked at Briot/Bouquet's book, but seems to have been lost in recent sources.)
Thank you!
Very interesting, thanks for sharing and doing such thorough research
I like where this is going, can't wait to see your take on contour integration, Cauchy's theorem and so on. Thank you for another great lecture!
when i was first learning this a few months back, the cauchy riemann equations were taught completely differently. However when you started off with the definition of real differentiability being whether or not you are able to approximate the function linearly made them feel so natural, like they just pop out, it's so nice
It's fascinating that the only reason why complex functions are infinitely differentiable if they're differentiable once, unlike differentiable functions from R^2 to R^2, is due entirely to complex multiplication. That's it. It gives us the CR-equations, and then everything else follows.
well, yes
giving the algebraic structure of a field makes things much more restrict
That's half the reason. The other half is that the functions are defined over a plane, and you can't have nondifferentiability when you are surrounded by complex differentiability unless you blow up. To see that this is essential, take any real valued function f(x) which is continuously differentiable once, but nowhere differentiable twice (integrate a nowhere differentiable function), and extend it linearly away from the real axis by defining f(x+iy) = f(x) + i f'(x) y. This function obviously isn't going to be holomorphic on any open set, but it is once complex differentiable on the real axis.
@@annaclarafenyo8185 well, yes. butI think they were referring to that relative to differentiability over R^2.
@@annaclarafenyo8185 also, Id make the distinction of being complex derivable over the real axis rather than differentiable
@@98danielray It's complex differentiable at all the points on the real axis. By the usual definition.
The best thing about Sir Richards is he knows these things in very depth and for him these topics are simple but he still teaches with such enthusiasm and excitement.
That, and he pronounces "z" correctly despite his English accent.
Looking forward to the rest of the lecture series.
wow, when I learned the notation df/dx never bothered to question it, and because of that it seemed some properties were just thrown there, but with a lim makes so much sense, thanks for sharing so much :)
Maybe it is called holomorphic - "holo" -> entire and "morph" -> shape i.e the shape remains entire after the transformation by such functions - which is kind of true as these functions locally preserve angles when their derivatives does not vanish.
This is how I was taught the name as well :D
In general, satisfying the Cauchy-Riemann equations is not sufficient for complex differentiability. If the derivatives of the real and imaginary parts are continuous, however, then it should be the case.
yeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
I thought the reason it was called holomorphic was because it was comparable to holographic
Thank you so much for this fabulous lecture! And also, if possible, I would very much like to see "topological or integration free" proof of the the fact that "derivative of holomorphic fuction is continous". Ahlfors said at the begining of chapter 4 in his book that such a proof has been found, but doen't present it in the book. And I also don't see such a proof in other complex analysis textbook. Thank you so much again!
I think the examples at the very end of the video are not correct. For instance, x and y are certainly not holomorphic as they are Re(z) and Im(z).
Some examples there could be z or z^2 = x^2-y^2+2xyi, ...
Am I right? or I'm missing something??
I think what is meant is either of the two functions: f(z) = Re(z), or the function, f(z) = Im (z). Input ANY complex variable and the function truncates either the real part or the imaginary part and outputs what remains. This type of truncating function would not be holomorphic.
Here he's specifying a single function, f, from the complex plane into the real numbers, essentially defining a function from the complex numbers into the real axis. He's using "holomorphic" as "which polynomials can be considered as real parts of a holomorphic function?"
Great lecture, is a pleasure to hear a Professor talking about Complex Analysis and address the fact that a Holomorphic function is not exactly equal to Analytic function. Great UA-cam channel.
This is my favorite math channel to listen to in headphones while at work. Richard Brocherds is a great expositor, and unlike a lot of lecturers does a great job speaking mathematics, and making the ideas on the board audible. It is difficult (at least for me) to learn mathematics "by ear", but I am very grateful for these videos as impressively clear and followable examples of mathematics teaching in the medium of audio lectures (a technique introduced by Vivienne Malone Mayes, one of the first Black woman math PHDs).
i love the internet. i have been learning from a fields medalist !
3.10 is an epiphany to me. Never did I get it, but learned to use it.
In the case of a real variable I still dont understand how it makes sense to split up the symbol df/dx when using, say, 'integration by parts'. Would the limit part follow the bottom half of df/dx around for it to make sense?
8:42 why does the matrix for A take that form? :(
simply take A as a complex number, multiply that with z and expand that out
three lines for defining/definitional equalities
FTW
idk how i got here but i'm so glad i found those before my exam
I keep forgetting what holomorphic means :(
Why should dz/dz(bar) be 0? z(bar) is not constant with respect to z.
As Richard points out, this is not actually what it means and is just shorthand notation for the partial derivatives with respect to x and y which he wrote down. Writing d/d(z bar) and z in terms of x and y you will get the result. It's slightly dangerous notation because it can lead you astray, but once you get used to it it can be a handy way of seeing at a glance whether a function is holomorphic or not.
Amazing lecture!
Can anyone explain at 9:30 why A(x+iy) =ReA*x-IMA*y +i(ImA*x+ReA*y)? Thank you very much.
A is in C, so when you write A= Re(A)+i*Im(A) and multiply both terms you get that
3:02 I think the standard notation for difference is \Delta. The notation d itself means "limit of delta"... Basically all textbooks of physics I know use such convention...
The word "holomorphic" was introduced by two of Cauchy's students, Briot (1817-1882) and Bouquet (1819-1895), and derives from the Greek ὅλος (holos) meaning "entire", and μορφή (morphē) meaning "form" or "appearance".[10]
The person who wrote this on Wikipedia has seen this video, and took it onto themselves to research it. [top comment]
This was great and well interlinked to what i already knew. Thank you
Derivatives have absolutely nothing to do w finance and hedge funds, it just means differentiation of a cmplx function