Complex Analysis -17: Cauchy's Theorem for Star-shaped Domains

Let U be an open start-shaped set. Let F be holomorphic on U. The the path integral of f over any closed path in U is zero. This is the most fundamental result of Cauchy theory. As a consequence, we show that any holomorphic function has local primitives,

Timestamp provided by Ishwarya and Priyanka Vaideesh.
00:00 Introduction
1:22 Proposition- Let U be a Star Shaped set & let f: U to C be holomorphic, Then f has a primitive in U
10:46 Reason for Why we have to choose the domain as Star-shaped?
18:29 Cauchy's Theorem for Star Shaped domain
21:21 Corollary- Let U be an open set in C & let f: U to C be holomorphic, then f has a local primitive.
23:24 Fundamental theorem of Algebra
40:27 Conclusion

[jahirul]

You are a living legend of mathematics sir, Thank a lot for such nice lectures...

[mathematicia]

I am not a person who trusts in God. But if there is any god for me then surely you are the one who is making my mathematics more pure day by day.