I have done this by using Laplace equation as I assumed $\log(|f(z)|)$ to be some variable '$w$' and proved the (second derivative of '$w$' with respect to $u$) +(second derivative of '$w$' with respect to $v$) is zero. Was it correct.
If $f(z)=u(x, y) +i v(x, y)$ is analytic then show that $\log(|f(z)|)$ is Zero
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complex-analysis
2 Answers
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If $f$ is holomorphic on some open set $D$ in $\mathbb C$ and if $f$ has no zeroes in $D$, then $\log(|f|)$ is harmonic in $D$.
Show your proof, then we will see whether your proof is correct.
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0I have added the picture of the proof which I have done. Please let me know whether I am right – 2017-02-11
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$f(z)=e^z$ is analytic and $\log|f|=x\neq0$.