Complex Analysis #4 | Complex line integrals

This is the fourth video in my "Essential Extract" series on Complex analysis.

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What this video covers

Intro (0:00)

1. Complex line integrals (0:26)
Definition
Example: z^n around the unit circle (2:08)

2. The Cauchy Goursat theorem (3:40)
Statement
Positive orientation (5:08)
Proof of the Cauchy Goursat theorem (5:40)

3. The Cauchy integral formula (6:56)
Statement
Proof of the Cauchy integral formula (7:42)

4. Morera's theorem (9:24)
Statement
Example 1: The Gamma function (10:27)
Example 2: The Riemann zeta function (12:40)
Another consequence of Morera's theorem (15:18)
Uniform convergence (15:59)

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What is covered in this series of videos

Cauchy-Riemann equations (video #3)
Complex line integrals (video #4 - this video)
Cauchy-Goursat theorem (video #4)
Cauchy integral formula (video #4)
Morera's theorem (video #4)
Laurent series (video #5)
Residue theorem (video #5)
Open mapping theorem (video #6)
Maximum modulus principle (video #6)
Analytic continuation (video #6)
Uniqueness theorem (video #6)
Argument principle (video #6)
Rouche's theorem (video #6)

We are going to move fast- I am going to try to cover all of this in under 2 hours. (Update: ended up finishing everything in just under 1 hour and 43 minutes)

There are a few notable omissions in this series:

Harmonic functions
Riemann surfaces
Inverse function theorem
Weierstrass products

However, if you understand everything else I am presenting, you should have no problem learning these topics quickly on your own.

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Prerequisites

The only real prerequisite is a solid understanding of single and multivariable calculus. However, if you have not seen complex analysis before, or at least taken some advanced math courses, it may be difficult to keep up. Still though, you might learn something at least from the first couple of videos.