Show that if is analytic within an on a simple closed contour C and is not on C, then

I've broken this down into two cases: is interior to C and is exterior to C

Case 1:

Since is interior to C we may apply the Cauchy Integral Formula so

and

and obviously:

However I have no idea what to do when is exterior to C. We're only given that f is analytic in the closure of C. I was hoping to use contour deformation or the Cauchy-Goursat Theorem but i don't know how.