Limits on the integral seem slightly mistaken at 18:26. Just below the "bubble" should be phi=-pi+epsilon, just above the bubble should be phi=pi-epsilon. Thus the signs of the epsilons should be opposite of what is used in the video. While this does not affect the ultimate result in practice, the following evaluation is in principle wrong as the path crosses the "bubble" twice with the phis used in the video. Also, upon evaluating the primitive, there is an additional small error of not including the 'i's for the epsilon. Correct would be 2*(pi-epsilon)*i prior to taking the limit. And this also makes sense, as we have taken slightly less than the complete 2*pi trip around the circle.
It's worth noting that you can have a primitive for 1/z with a branch cut anywhere you'd like (connecting 0 and infinity), but you can't have one without any branch cut. So there's no problem integrating along -1+it from t=-1 to 1 with the FToC if you just have your arguments go 0 to 2pi, but you can't do any circle around the origin without a hack.
Great lecture, please keep up the good work! Being a student who didn’t take a complex analysis course at my university, I’m really happy to find such content on TH-cam.
?? For me as a former math. student at least one complex analysis course was mandatory, normally even a 2nd one (complex analysis II & Riemann surfaces) is usual for a master degree. (You can't understand analytic number theory without complex analysis) I agree with you: Michael's videos are good and inspiring work, that'll help you understanding several maths. sections.
@@scp3178 I’m doing a degree in applied mathematics and we have two specializations here in Ukraine: informatics and applied maths itself. I’d picked the first one, and that’s why some of the common math courses in my program were replaced with various CS courses such as algorithm analysis, cryptography etc. Though complex analysis is perhaps the only major “non pure” math branch I’ve missed.
Same here - got my master's in math but never took a complex analysis course since I was more focused on discrete stuff and CS. Very nice to have these videos to learn some complex analysis after the fact :)
@@jimskea224 Between Brown, Churchill, and Dr Penn, you can't go wrong. I have a foreign language disability and so haven't read the other text. Thanks.
at 7:43 you assume without justification (although I suppose it is 'intuitively obvious') that int_{gamma_1 + gamma_2} f(z) dz = int_{gamma_1} f(z) dz + int_{gamma_2} f(z) dz. To compute the integral without this assumption is not that bad; one just has to calculate the explicit parameterization of gamma_1 + gamma_2 (which on [0, 1/2] is gamma_1 at double speed and on [1/2, 1] is gamma_2 at double speed and delayed to be on the interval [1/2, 1])
At 18:00 to avoid the problematic point, should not the boundaries be like exp^(-i(Pi+epsilon)) to exp^(i(Pi-epsilon)) instead of int(1/z, exp^(-i(Pi-epsilon)), exp^(i(Pi+epsilon)))?
Just a small note, when he takes the limit as epsilon goes to zero, it should be from the left, not right. Otherwise it actually crosses the point he's trying to omit. The result is the same, however.
Nice lecture, but it would have been nice if you gave an actual definition of the integral over contour lines. Since there is no definition some of the steps in your calculations feel a bit handwavy at times
At the very beginning, you said that a "domain" is a simply-connected open set (and you needed that for the given version of Cauchy's Theorem). I know of at least one textbook that would just say "connected" open set. Certainly a connected open set contains a simply-connected open set around each point, but is it more common for "domain" to mean "simply-connected"?
My default definition of "domain"/"region" is non-empty + connected + open, the convention used in PDEs. I would say that the responsibility lies on the author to always make sure that he/she follows-up what is precisely meant by "domain", as part of good math writing and communication, just like how it is done here.
@@schweinmachtbree1013 what definition of simply-connected do you have in mind? In the definition I'm familiar with, the punctured plane or an annulus would be connected yet not simply-connected.
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Limits on the integral seem slightly mistaken at 18:26. Just below the "bubble" should be phi=-pi+epsilon, just above the bubble should be phi=pi-epsilon. Thus the signs of the epsilons should be opposite of what is used in the video. While this does not affect the ultimate result in practice, the following evaluation is in principle wrong as the path crosses the "bubble" twice with the phis used in the video. Also, upon evaluating the primitive, there is an additional small error of not including the 'i's for the epsilon. Correct would be 2*(pi-epsilon)*i prior to taking the limit. And this also makes sense, as we have taken slightly less than the complete 2*pi trip around the circle.
Fun fact: if you integrate zbar counterclockwise around any simple loop, you get 2i times the area inside the loop!
5:49 The integral should be on t*dt not t^2*dt. The result is correct.
6:00 small error: the integral is integral[t, {t,0,1}] not t^2
I am only halfway though this video and this is the best introduction to integration in complex analysis I have ever seen.
It's worth noting that you can have a primitive for 1/z with a branch cut anywhere you'd like (connecting 0 and infinity), but you can't have one without any branch cut. So there's no problem integrating along -1+it from t=-1 to 1 with the FToC if you just have your arguments go 0 to 2pi, but you can't do any circle around the origin without a hack.
Great lecture, please keep up the good work!
Being a student who didn’t take a complex analysis course at my university, I’m really happy to find such content on TH-cam.
?? For me as a former math. student at least one complex analysis course was mandatory, normally even a 2nd one (complex analysis II & Riemann surfaces) is usual for a master degree. (You can't understand analytic number theory without complex analysis)
I agree with you: Michael's videos are good and inspiring work, that'll help you understanding several maths. sections.
@@scp3178 I’m doing a degree in applied mathematics and we have two specializations here in Ukraine: informatics and applied maths itself. I’d picked the first one, and that’s why some of the common math courses in my program were replaced with various CS courses such as algorithm analysis, cryptography etc. Though complex analysis is perhaps the only major “non pure” math branch I’ve missed.
Same here - got my master's in math but never took a complex analysis course since I was more focused on discrete stuff and CS. Very nice to have these videos to learn some complex analysis after the fact :)
6:07 PENNultimate step yassss
(please tell me this was an Easter egg in response to my comment on the weights/partitions video!)
18:30 The limits on the integral should be e^i(pi-epsilon) and e^i(-pi+epsilon). Again, the results are correct.
Exactly!
I wish I had these videos when I took Complex Variables. Thank you for these!
Are you planting to do a video on conformal maps on polygons?
We used Churchill for complex analysis in the mid 1970s and it was an excellent textbook. Is this still in print? Or is there a newer text being used?
Damn Winston Churchill wrote complex analysis books on top of running the UK? nice
I use both (Brown &) Churchill and Avila's "Variáveis Complexas e Aplicações" for my course in Complex Analysis.
@@jimskea224 Between Brown, Churchill, and Dr Penn, you can't go wrong. I have a foreign language disability and so haven't read the other text. Thanks.
Finally, thank you. Hope you make more videos regarding Complex Integration.
at 7:43 you assume without justification (although I suppose it is 'intuitively obvious') that int_{gamma_1 + gamma_2} f(z) dz = int_{gamma_1} f(z) dz + int_{gamma_2} f(z) dz. To compute the integral without this assumption is not that bad; one just has to calculate the explicit parameterization of gamma_1 + gamma_2 (which on [0, 1/2] is gamma_1 at double speed and on [1/2, 1] is gamma_2 at double speed and delayed to be on the interval [1/2, 1])
At 18:00 to avoid the problematic point, should not the boundaries be like exp^(-i(Pi+epsilon)) to exp^(i(Pi-epsilon)) instead of int(1/z, exp^(-i(Pi-epsilon)), exp^(i(Pi+epsilon)))?
Just a small note, when he takes the limit as epsilon goes to zero, it should be from the left, not right. Otherwise it actually crosses the point he's trying to omit. The result is the same, however.
At 06:00, how did you get (1+i)-squared / 2. Where did the / 2 come from ?
18:20 minor sign error, it should be -pi+epsilon and pi - epsilon
Guys is there a way to find the solution of exercises that the professor give us ?
i think that analyticty needs the continuity of first partial derivative of u(x,y) and v(x,y) over x and y
Complex integration? More like "Your videos are stoking my imagination." Thanks so much for making and sharing them!
Nice lecture, but it would have been nice if you gave an actual definition of the integral over contour lines. Since there is no definition some of the steps in your calculations feel a bit handwavy at times
Wow this is hoe math must de done congrats
Next up would be Laurent series?
I imagine Cauchy Integral Formula comes next
i can't find a good proof of green's theorem anywhere
At the very beginning, you said that a "domain" is a simply-connected open set (and you needed that for the given version of Cauchy's Theorem). I know of at least one textbook that would just say "connected" open set. Certainly a connected open set contains a simply-connected open set around each point, but is it more common for "domain" to mean "simply-connected"?
My default definition of "domain"/"region" is non-empty + connected + open, the convention used in PDEs. I would say that the responsibility lies on the author to always make sure that he/she follows-up what is precisely meant by "domain", as part of good math writing and communication, just like how it is done here.
for subsets of *C* connectedness and simply-connectedness are equivalent so it doesn't matter ( the same is true for subsets of *R* ^ _n_ )
@@schweinmachtbree1013 what definition of simply-connected do you have in mind? In the definition I'm familiar with, the punctured plane or an annulus would be connected yet not simply-connected.
@@diribigal oh silly me, it's connectedness and path-connectedness that are equivalent
man you move too fast its kindah hard to follow up