Week5Lecture3: The fundamental theorem of calculus for analytic functions

[@Kybeline]

The best complex analysis (either playlist, pdf-script or book) I've ever met! I'm an absolutely beginner, but I've mangaged it so far and I'm confident now that I can make it to the end. Thanks a lot for putting it online.

[@mnada72]

Where are these pdf or book ?

[@randomvideos-qz1fl]

You are a godsend❤❤

[@oystein98]

For a layman with some common sense: If we think of the "F" as a function measuring some kind of height in the terrain, and the "f" as giving the slope of the terrain, then the integral of f around any curve is the net sum of uphills and downhills. Therefore it's easy to see that the net sum of uphills and downhills, (the integral,) is zero if we come back to the starting point. It doesn't matter which route we take. If we split the trip it's also easy to see that the net sum depends only of the endpoints.

[@trevorsimpson8788]

Prof, I'm looking for the fundamental theorem of Algebra proof in your series.

[@leonig100]

These are very good lectures on what I regard as a very difficult subject to get across. Thank you. My problem is that in the examples of picture 3 and 4 the path seems to be well defined but in the example of picture 5 the path can be anything between the limits. Where am I going wrong. Another problem that I have is what is the difference between the path integral and the arc length you described in the previous lecture?

[@petrataylor8628]

You are absolutely correct! The point here is that if f is continuous and has an antiderivative (aka primitive) in the entire domain where the path is located, then the integral only depends on the values of f at the initial and terminal points of the path, but not how the path got from its initial to its terminal point. When it comes to arc length, there is no function f that we integrate; rather, arc length is a property of the path itself (and there is that absolute value sign around the derivative that we need to remember when computing arc length).