Week5Lecture4: Cauchy's Theorem and Integral Formula

[@cyclol]

Thank you for this lecture series. Most complex analysis lecture series online consist of rigorous proof which does not help on homework! Your lightweight proof idea combined with many examples is extremely helpful!

[@christianorlandosilvaforer3451]

thats true .... she is so clean in the explanations! i love her!

[@Cashman9111]

I don't get one thing : why doesn't Cauchy's Theorem follows immadietely from fundamental theorem of calculus ? If we have closed curve then obviously in simply connected domain F(x) - F(x) = 0 for any x ?!

[@Asdfghjkl-w3j]

maam you have explained it the best possible way. thankx a lot. you are doing a very noble thing here.

[@christianorlandosilvaforer3451]

thanks a lot off for your time to make this lectures, i have watched from lec 1 to this and i have learned all, becouse your explanations are so clean and simple! great teacher!

[@dianampm99]

Thank you so much for these videos. I've probably rewatched this one a couple of times, and its so well explained. So good right now, since my university online classes have been so terrible during quarantine.

[@mpandelithelma3985]

Thank you for making this concept easy to understand

[@OrionKalas]

amazing examples, thank you!

[@jackstacks3989]

Wonderful series bravo and thanks!

[@tubayilmaz4650]

At 6:45 how does the orientation of gamma 2 becomes clockwise?

[@Cashman9111]

why doesn't this theorem follow immediately from the path independence theorem ? F(a)-F(b), since a=b so obviously it's zero... and at 5:15, why do we have to prove that ? these integrals equal zero, so of course they equal each other...

[@abhinovenagarajan.s7237]

I might be wrong, but the path independence theorem in the previous lecture (I assume?) assumes that the integrand can be written as a derivative of some function. This theorem is valid for any integrand as long as it is analytic in region enclosed by that closed curve.

[@georgeobrien1011]

@@abhinovenagarajan.s7237 yes, and in the discussion on slide 2 she mentions specifically that exp(z**3) has no anti-derivative, but is analytic

[@ilfakt]

A bit misleading for me at 10:39 because you introduced cauchy's theorem for simply connected domains. Obvious this is not the case in that example when you chose D.

[@PetraBonfertTaylor]

The example you are stating uses the Corollary introduced at 4:47, which only requires a function that is analytic in a domain that contains both curves (which are simple, closed, one inside the other, oriented counterclockwise) as well as the region between the two curves.

[@ilfakt]

Danke Frau Prof. Bonfert-Taylor!