Let a function f(z) be analytic in a domain D. (a) Prove that if f(z)? is also analytic in D, then f(z) is constant in D. [Hint: Use the Cauchy-Riemann equations.] (b) Prove that if |f(z)| is constant throughout D, then f(z) must be constant throughout D. [Hint: |f(z)|² = f(z)f(z)?.]
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We are given that $f(z)$ is analytic in domain $D$. This means that $f(z)$ is differentiable in $D$ and satisfies the Cauchy-Riemann equations. Show more…
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