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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 + re^{iθ}. Substituting into 1) we get

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