Three questions to do with Cauchy's integral formula are evaluated in clear, easy to understand steps.
1. Show that
where C is the semicircule |x|=R>1, lm(x)>=0 from x=-R to z=R.
2. Use Cauchy's integral formula for derivatives to evaluate
where |z-2|=2 is oriented clockwise.
where C is the square with vertices 2+/-2i, -2+/-2i, oriented counterclockwise
This solution reviews some basic properties of line integration, including the Cauchy integration formula for derivatives. Then a detailed solution is presented for each of the three questions to show by example how to apply these properties.