Complex analysis: Integration

This lecture is part of an online undergraduate course on complex analysis.

We define integration of a complex function along a path as the limit of a sum, and give its basic properties. We finish by calculating the integral of z^n around a circle.

[Juan-yjnn]

You're helping a lot of people with this!

[Yakushii]

Thank you so much for these lectures. They are an absolute godsend!

[dneary]

Hi Prof. Borcherds! If you are looking for ideas for future series, I for one would love an accessible introduction to elliptical curves, particularly for cryptography and in the context of non-linear diophantine equations.

[sebastiansanfunas]

This video is great, your series in complex analysis is just the best, it’s helped me greatly to understand these basic theorems on a deeper level, thanks!

[darkflower]

Thank you very much for making these videos public sir! Was devastated by the complex analysis course I took in my undergrad and am very interested in properly learning the subject. Will be following along this series!

[arshadali]

Good that REB spent some time on the differing (but equivalent) ways of defining the complex integral. In my opinion, the older writers do it best -- principally Titchmarsh in "Theory of Functions" and Copson in "Theory of Functions of a Complex Variable." Stewart and Tall also spend some effort on it, defining it in terms of the Riemann-Stieltjes integral.

[ablclanmarazm]

19:35 I’m sure that Prof. Borcherds was joking but I’ll explain if someone wants to know. Because we choose i to be in the upper half-plane we get the complex numbers rotated 90 degrees counter-clockwise when multiplying by i. If you plot exp(ix) series you will get the counter-clockwise spiral because of this. Seems good enough reason to me

[lhcoco]

Well, explained. Thanks you so much, it helped enormously !

[davidaugustyn]

Should I already know group theory before this course

[Someone-crcj]

7:03 greatest integral sign i have ever seen.

[hewafaris]

Is there exists any type of double integral in complex analysis? Please answer. With regards

[hewafaris]

Hello sir. Could you please help me? I wonder if you can answer me, why don't we have double integral in complex analysis. With regards.

[serioussearch]

The complex integral definition is reminiscent of the real arc length approximation by linear segments.

[TheLolle]

I can‘t even make out what is written in that yellow ink 😅

[scoobydoo]

I will never understand why mathematicians are so keen on saying that things are obvious. Even if one thinks that something should be obvious, it doesn't do much for a struggling student to tell them that something is obvious. In fact, I think it can be quite frustrating. I sometimes wonder if it's so that mathematicians can inflate their own ego. If not, I can't really see why

[hyperduality]

Integration is dual to differentiation.
Infimum is dual to supremum.
Points are dual to lines -- the principle of duality in geometry.
Convergence (integration, syntropy) is dual to divergence (differentiation, entropy).
Reductionism is dual to holism.
Limits, boundaries, barriers = duality.
"Always two there are" -- Yoda.

[hannesstark]

I love your lectures, but I think a microphone with less distortion might be an easy and worthwhile investment.

[eamonnsiocain]

“I don’t want you actually to read this . . . “

[bigchungus]

i thought this was a 4chan greentext from the thumbnail