Use the proposition in Problem 1 to show that if f and g are analytic and f′(z) = g′(z), t...

Use the proposition in Problem 1 to show that if f and g are analytic and f′(z) = g′(z), then f(z) = g(z) + c, where c is a constant. [Hint: Form h(z) = f(z) − g(z).]

Problem 1

In this problem you are guided through the start of the proof of the proposition:

Proof We begin with the hypothesis that f is analytic in D and hence it is differentiable throughout D. Hence by (1) and the assumption that f′(z) = 0 in D, we have . Now complete the proof.

 (1)