28:00 The wikipedia article motivates the the form of a residue at infinity by claiming that residues are taken of "differential forms" not functions. I think you've spoke on this topic in the past but not in this context. I hope future videos will clear this point up.
Another excellent video on complex analysis! Thank you for the awesome lessons/content! Can we also expect to see a course on differential geometry? I know I'd love to see one. :)
If I encounter a big partial fraction decomposition problem which requires building up a big power, I'll use the residue theorem to help evaluate it. It beats having to do a ton of algebra to solve a big system of equations. For example, to find C in some C/x^3 term, I would multiply both sides of the equation by x^2 then apply the residue theorem to both sides of the equation. C is then equal to the residue of the other side of the equation at 0. You are still required to calculate repeated derivatives of something, though. Additionally, if you have a term like (E*x+F)/(x^2+1), you can actually use complex analysis to factor the denominator further and split it into two terms as E/(x+i) + F/(x-i) instead. Then once again you can apply the residue theorem for each E and F.
This series of lectures is a great idea. After the basic core of Complex Analysis, which pretty much done, will you do any videos after? And will they be more applied (transforms, Fourier series, etc) or more theoretical (Riemann mapping theorem, Mittag-Leffler's theorem, etc)?
These are supporting a class he's teaching spring of 2022, so I assume future series will depend on what he teaches next, and possibly on the class format his school is using at the time.
Imagine if I want to calculate the residue of the function f(z)=1/z^2 at infinity. With that formula, both z^2 cancel out and turn out in a constant (-1), which is analytic at zero. What should I do?
I think he did that because sinh z would make it a 2nd order pole instead of a 3rd order. One factor of z would be removable at 0. Taking the series expansion of sinh z at 0 would show it.
They're first order because their derivatives aren't also zero (or the limit of (sin z)/z isn't 0). The essential singularity of sin 1/z at 0 is related how sin doesn't have a limit (even infinity) as z goes to infinity, and it's the "graph looks solid" situation in real analysis, rather than "graph leaves the page". There's not really an "essential" category of zeros corresponding to essential singularities in the way in which zeros and poles correspond.
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At 12:00 Shouldn't the derivative of q have a q(z) minus q(znaught) in there somewhere rather than just q(znaught) ?
He should've mentioned that because q(z0)=0 then you can just turn q(z) into q(z)-q(z0) without changing it's value, since you're subtracting zero.
28:00 The wikipedia article motivates the the form of a residue at infinity by claiming that residues are taken of "differential forms" not functions. I think you've spoke on this topic in the past but not in this context. I hope future videos will clear this point up.
Another excellent video on complex analysis!
Thank you for the awesome lessons/content!
Can we also expect to see a course on differential geometry?
I know I'd love to see one.
:)
If I encounter a big partial fraction decomposition problem which requires building up a big power, I'll use the residue theorem to help evaluate it. It beats having to do a ton of algebra to solve a big system of equations. For example, to find C in some C/x^3 term, I would multiply both sides of the equation by x^2 then apply the residue theorem to both sides of the equation. C is then equal to the residue of the other side of the equation at 0. You are still required to calculate repeated derivatives of something, though.
Additionally, if you have a term like (E*x+F)/(x^2+1), you can actually use complex analysis to factor the denominator further and split it into two terms as E/(x+i) + F/(x-i) instead. Then once again you can apply the residue theorem for each E and F.
thank you so much for all your videos!
These videos a phenomenal, thank you!
Thank you for your videos. It always helpful
This series of lectures is a great idea. After the basic core of Complex Analysis, which pretty much done, will you do any videos after? And will they be more applied (transforms, Fourier series, etc) or more theoretical (Riemann mapping theorem, Mittag-Leffler's theorem, etc)?
These are supporting a class he's teaching spring of 2022, so I assume future series will depend on what he teaches next, and possibly on the class format his school is using at the time.
Around 18:10 : isn’t there a factor of 2 missing in front of the a-1 coefficient?
Could you plz prepare a series on real analysis the same way this series is? It's really awesome and compact.
Imagine if I want to calculate the residue of the function f(z)=1/z^2 at infinity. With that formula, both z^2 cancel out and turn out in a constant (-1), which is analytic at zero. What should I do?
Then the residue is zero
Linear algebra!
At 23:46 a sinh example disappears and is replaced by a cosh example.
I think he did that because sinh z would make it a 2nd order pole instead of a 3rd order. One factor of z would be removable at 0. Taking the series expansion of sinh z at 0 would show it.
silly question. why are the zeroes of sine and cosine first order and not essential?
They're first order because their derivatives aren't also zero (or the limit of (sin z)/z isn't 0). The essential singularity of sin 1/z at 0 is related how sin doesn't have a limit (even infinity) as z goes to infinity, and it's the "graph looks solid" situation in real analysis, rather than "graph leaves the page". There's not really an "essential" category of zeros corresponding to essential singularities in the way in which zeros and poles correspond.
Books?
I'm more of a physicist so I recommend complex variables and applications by Churchill.
If you search "robert b ash complex analysis" you'll find his website with a free version of a good book that was published by Dover.
I'm fairly sure he's following Gamelin for these lectures