Prove No Analytic Function F on Annulus D: 1<|z|<2

[oab729]

Homework Statement

Prove that there does not exist an analytic function on the annulus D: 1<|z|<2, s.t. F'(z) = 1/z for all z in D.

Homework Equations

The Attempt at a Solution

Assume F exists, then for z in D, not a negative number, F(z) = Log z + c since Log' z = 1/z... Lost

 

[Billy Bob]

Find a path C such that $$\int_C \frac1z\,dz\neq0$$