Nitpick about 22:30 (the second of the three equivalent statements): g isn't defined at z_0. Shouldn't we instead say that g has a removable singularity at z_0?
I just wanted to say that these course videos have been awesome. I haven't been able to keep up super well with the material during my term, but now that the summer is rolling around... :)
I'm a bit confused on the definition of isolated singularity. Wouldn't that make any point in an analytic function be an isolated singularity? Because, if a function is analytic and we just consider the point z_0 ( even if f is analytic on z_0), we can definitely find a punctured disk where f is also analytic. Or am I missing something? Does the fact that we are considering a punctured disk mean that we don't really care whether f is analytic on z_0 or not?
To clarify my question. Take for example f(z)=z^2 and take the point z=0. Is 0 an isolated singularity? Since f(z) is analytic everywhere, it must also be analytic on the punctured disk of unit 1 from origin because we just removed the point 0 from the disk and it is obviously analytic in the other points from there? Does this also mean that if a function is analytic everywhere then every point is isolated singularity?
@@jimallysonnevado3973 That is more or less correct. The correct phrasing is: z0=0 is a removable singularity of the function f:C\{0} -> C, f(z) = z^2... So if you were tasked to extend f to an analytic function from C->C, you will set f(0) = 0. But this is slightly cheating, because we started with C->C and deleted the point z0=0. . In the business of isolated singularities, you should really keep in mind a different setup: You are first given f:D\{z0}->C, i.e. f is only defined on a punctured disc D\{z0}. We call z0 a singularity because f is undefined at z0. f(z) = complicated. You are then asked to identify what kind of isolated sing is z0. If z0 turns out to be a removable sing, you are further tasked to extend f to D->C.
Ohh, I got it already. The way he defined f, implicitly stated that the domain of f is the punctured disk. My confusion arosed because what I thought was that f has some domain where it is analytic (This domain is potentially bigger than the punctured disk.) and the punctured disk is only inteded to be a subset of this domain. My understanding is that if you can find a subset of the domain which is a punctured disk around some z_0 where it is analytic then z_0 is an isolated singularity.
Around 28:00 The preconditions of the theorem is maybe not enough. It has to be an isolated singularity which is not removable. Otherwise the limit doesn't need to be infinity (?)
The theorem says that if the isolated singularity z_0 is a pole then the limit is infinity. Here 'pole' means pole of positive order so a 'pole' is by definition not a removable singularity. (removable singularities can be thought of as "poles of order 0", but that is not being considered here)
Convergence is dual to divergence. Poles (eigenvalues) are dual to zeroes -- optimized control theory. Analytic (mathematics) is dual to synthetic (physics) -- Immanuel Kant. "Always two there are" -- Yoda.
@@focusmaestro4013 Symmetric matrices (real eigenvalues) are dual to anti-symmetric matrices (complex eigenvalues) -- linear algebra. Real numbers are dual to complex numbers -- complex numbers are dual. Bosons or symmetric wave functions are dual to Fermions or anti-symmetric wave functions -- spin statistics theorem. Bosons are dual to Fermions -- atomic duality or wave/particle duality. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy). Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! The 4th law of thermodynamics is hardwired into mathematics and mathematical thinking. Duality creates reality! Teleological physics (syntropy) is dual to non-teleological physics (entropy).
![The most beautiful equation in math, explained visually [Euler’s Formula]](/img/n.gif)
The limit at 21:21 should be r->0, not r->infinity. Then M times a positive power of r (for negative n) goes to zero as required.
You were few minutes quicker than my post 😉
Agreed, r was chosen to be less than R, so it doesn't make sense if it surpasses R towards infinity.
I was stuck in this part of the video, I went to the comments to see if anyone else had found the error
Thanks, got me confused for a second
Nitpick about 22:30 (the second of the three equivalent statements): g isn't defined at z_0. Shouldn't we instead say that g has a removable singularity at z_0?
I just wanted to say that these course videos have been awesome. I haven't been able to keep up super well with the material during my term, but now that the summer is rolling around...
:)
I hope we have a video on common meromorphic function examples. Not like exponential or trig functions, but weird functions like the Gamma function
21:40 r is less than R, so it shouldn't go to infinity. For n
Thank you professor great work
I'm a bit confused on the definition of isolated singularity. Wouldn't that make any point in an analytic function be an isolated singularity? Because, if a function is analytic and we just consider the point z_0 ( even if f is analytic on z_0), we can definitely find a punctured disk where f is also analytic. Or am I missing something? Does the fact that we are considering a punctured disk mean that we don't really care whether f is analytic on z_0 or not?
To clarify my question. Take for example f(z)=z^2 and take the point z=0. Is 0 an isolated singularity? Since f(z) is analytic everywhere, it must also be analytic on the punctured disk of unit 1 from origin because we just removed the point 0 from the disk and it is obviously analytic in the other points from there? Does this also mean that if a function is analytic everywhere then every point is isolated singularity?
@@jimallysonnevado3973 That is more or less correct. The correct phrasing is: z0=0 is a removable singularity of the function f:C\{0} -> C, f(z) = z^2... So if you were tasked to extend f to an analytic function from C->C, you will set f(0) = 0. But this is slightly cheating, because we started with C->C and deleted the point z0=0.
.
In the business of isolated singularities, you should really keep in mind a different setup: You are first given f:D\{z0}->C, i.e. f is only defined on a punctured disc D\{z0}. We call z0 a singularity because f is undefined at z0. f(z) = complicated. You are then asked to identify what kind of isolated sing is z0. If z0 turns out to be a removable sing, you are further tasked to extend f to D->C.
Ohh, I got it already. The way he defined f, implicitly stated that the domain of f is the punctured disk. My confusion arosed because what I thought was that f has some domain where it is analytic (This domain is potentially bigger than the punctured disk.) and the punctured disk is only inteded to be a subset of this domain. My understanding is that if you can find a subset of the domain which is a punctured disk around some z_0 where it is analytic then z_0 is an isolated singularity.
I guess the assumption is that f doesn't exist at z0
@@jimallysonnevado3973 it has to be a singularity for it to be an isolated one. I had the same thought.
Riemann Zeta function here we come!
Around 28:00
The preconditions of the theorem is maybe not enough. It has to be an isolated singularity which is not removable. Otherwise the limit doesn't need to be infinity (?)
The theorem says that if the isolated singularity z_0 is a pole then the limit is infinity. Here 'pole' means pole of positive order so a 'pole' is by definition not a removable singularity.
(removable singularities can be thought of as "poles of order 0", but that is not being considered here)
30:32 It looks like Michael assumed that 1/|f(z)| has a isolated zero at z_0?
yes if f(z) has an isolated singularity at z_0 then 1/f(z) has an isolated zero at z_0, pretty obvious
Thank you for the awesome lessons/content!
Can we also expect to see a course on differential geometry?
I know I'd love to.
:)
Please abstract linear algebra
Convergence is dual to divergence.
Poles (eigenvalues) are dual to zeroes -- optimized control theory.
Analytic (mathematics) is dual to synthetic (physics) -- Immanuel Kant.
"Always two there are" -- Yoda.
ok sir, noted
@@focusmaestro4013 Symmetric matrices (real eigenvalues) are dual to anti-symmetric matrices (complex eigenvalues) -- linear algebra.
Real numbers are dual to complex numbers -- complex numbers are dual.
Bosons or symmetric wave functions are dual to Fermions or anti-symmetric wave functions -- spin statistics theorem.
Bosons are dual to Fermions -- atomic duality or wave/particle duality.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy).
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
The 4th law of thermodynamics is hardwired into mathematics and mathematical thinking.
Duality creates reality!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).