Prove: Gauss’ mean value theorem. Let f(z) be analytic inside and on a circle C of radius r and center at a. Then f(a) is the mean of the values of f(z) on C, i.e.
Proof. Using Cauchy’s integral formula
The equation of C is |z - a| = r or z = a + reiθ. Substituting into 1) we get