Let f(z) be an analytic function defined in a domain D. Show that f(z) must be constant if one of the following statements is true for all z ? D: (a) f(z) is real valued (b) Re(f(z)) or Im(f(z)) are constant functions (c) Re(f(z)) = [Im(f(z))]². HINT: Use the Cauchy Riemann equations.
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If $f(z)$ is real-valued, then $f(z) = u(z) + iv(z)$, where $v(z) = 0$ for all $z \in D$. So, $u(z)$ is a harmonic function and $v(z)$ is a constant function. Show more…
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