Prove: Let f(z) be analytic in a simply-connected region R. If a and b are any two points in R, then

is independent of the path in R joining a and b.

Proof. Let C₁ and C₂ be two curves running from point a to point b in R as shown in Fig. 1. Then by Cauchy’s theorem

or

Thus

Consequently

which is what we wish to prove.