Lecture slides: the-argument-principle.pdf

Reading suggestion: Good support material is Spanne's "blixtkurs i komplex integration" blixtint2.pdf. For background you might find this mathematical description useful (Spanne, S: Lineära system. KF-Sigma, 1997, pp.325-344). Another good source are the lecture notes "Complex Analysis" by Stein and Shakarchi (google to find a pdf!).

The argument principle is all about determining how much the argument of a function varies as the input moves along a contour. In this video we introduce the mathematical machinery used to quantify how much the argument of a function varies.

Armed with this method for calculating the change in the argument of a function, we are now ready to derive the argument principle. The argument principle shows that the number of times the output of a complex function winds around the origin when the input moves around a closed curve is determined by the number of zeros and poles of the function enclosed by the input curve.

If you want a bit more detail in deriving the argument principle, check out this video: