1. The next three calculations provide some insight into Cauchy’s theorem, which we treat in the next chapter. (a) Evaluate the integrals for all integers n. Here γ is any circle centered at the...

1. The next three calculations provide some insight into Cauchy’s theorem, which we treat in the next chapter.

(a) Evaluate the integrals

for all integers n. Here γ is any circle centered at the origin with the positive (counterclockwise) orientation.

(b) Same question as before, but with γ any circle not containing the origin.

(c) Show that if |a|

where γ denotes the circle centered at the origin, of radius r, with the positive orientation.

2. Suppose f is continuous in a region Ω. Prove that any two primitives of f (if they exist) differ by a constant.