exp(1/z) is the typical example in "Math Methods for Physics texts" of an essential singularity, then immediately avoided! It's very fun to see the consequences of these beasts.
Small correction. At a pole, it is not true that lim_{z->z_0}f(z) = infinty, rather lim_{z->z_0}|f(z)| = infinity. Indeed, the former already fails over R, where lim_{x->0}1/x does not exist since the left and right limits do not coincide. Great video btw!
Thanks! When we talk about infinity in the complex plane, aren't we are referring to the point at infinity on the riemann sphere? So we don't necessarily need absolute values as you mentioned
@@qncubed3 Agreed. Your statement is correct. In the complex plane, there is no meaningful notion of positive ∞ or negative ∞. So we simply say that lim 1/z (z -> 0) = ∞, where -∞ = i·∞ = ∞.
Great Picard doesn't just say that f(z) attains every value bar one in any neighbourhood of the essential singularity - it says it attains all those values *infinitely many times* in those neighbourhoods. Which is mad. I hope to understand its proof one day too lol.
I would guess, an essential? Rewriting cot in terms of complex exponentials and simplifying we see something of the form exp(1/exp) which would have infinitely many negative powers unless you're trying to trick me and it doesn't...
Little hard to say, since I've been studying bits and pieces over the past 3 years... Definitely focus on a particular area (eg: contour integration), and work with a lot of examples and get plenty of exposure. The past few weeks have been Mobius transformations for me (this video is part of that series!) so I've just been watching different videos on the same topics, browsing the internet for proofs of theorems, and stack exchange is always a good resource for problems, solutions and proofs :)
exp(1/z) is the typical example in "Math Methods for Physics texts" of an essential singularity, then immediately avoided!
It's very fun to see the consequences of these beasts.
Always pleasure to see your lectures . God bless you sir
Looking forward to see the coming content on this. As always, it was a very fun and interesting to watch!
New videos！Happy to see!
New video！Happy to see!
i saw that theorem as any neighborhood V of z_0 essential singularitie satisfies that f(V) is dense, pretty cool theorem ^^
Small correction. At a pole, it is not true that lim_{z->z_0}f(z) = infinty, rather lim_{z->z_0}|f(z)| = infinity. Indeed, the former already fails over R, where lim_{x->0}1/x does not exist since the left and right limits do not coincide. Great video btw!
Thanks! When we talk about infinity in the complex plane, aren't we are referring to the point at infinity on the riemann sphere? So we don't necessarily need absolute values as you mentioned
@@qncubed3 Agreed. Your statement is correct. In the complex plane, there is no meaningful notion of positive ∞ or negative ∞. So we simply say that lim 1/z (z -> 0) = ∞, where -∞ = i·∞ = ∞.
Great Picard doesn't just say that f(z) attains every value bar one in any neighbourhood of the essential singularity - it says it attains all those values *infinitely many times* in those neighbourhoods. Which is mad. I hope to understand its proof one day too lol.
Indeed, and little picard > liouville. Insane :O
Supporting science ...
THANK YOU
good video
First !! I'm stoked
:O
@@qncubed3 I did not understand A thing ☹️
great picard's theorem next??
😶🌫️
Which books do you like for an intro to complex analysis?
I haven't read any complex analysis books, however The Math Sorcerer has a few videos on that
Stein-Shakarchi - Complex Analysis owo
@@mfourier thanks.
What kind of singularity does cot(cot(z)) have at 0?
I would guess, an essential? Rewriting cot in terms of complex exponentials and simplifying we see something of the form exp(1/exp) which would have infinitely many negative powers unless you're trying to trick me and it doesn't...
@@qncubed3 I think that the singularity at 0 is not isolated, but an accumulation point. Think of 1/sin(1/z).
How did you find it best to self study complex analysis
Little hard to say, since I've been studying bits and pieces over the past 3 years... Definitely focus on a particular area (eg: contour integration), and work with a lot of examples and get plenty of exposure. The past few weeks have been Mobius transformations for me (this video is part of that series!) so I've just been watching different videos on the same topics, browsing the internet for proofs of theorems, and stack exchange is always a good resource for problems, solutions and proofs :)
q chingados que grado es este? seguro no estado poniedo antencion en mi clase de matematicas
Physics building always have empty rooms :)
definitely using these now