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Suppose that $f$ is analytic on some region $R\in\mathbb{C}$. If Im$(f)$ = $k\cdot$Re$(f)$ for some nonzero constant $k\in\mathbb{C}$, then show that $f$ is constant on $R$.

I know that if $f'(z)=0$ for all $z$ in some region $R$, then $f(z)$ is constant. However, I'm not sure how to apply this to the question at hand. I also think the Cauchy-Riemann equations could be helpful, but again, I'm not sure how to apply them to the question.

Any guidance would be greatly appreciated. Thank you!

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    $\begingroup$ The constant $k$ had better be a real number, since both $\text{Im}(f)$ and $\text{Re}(f)$ are real-valued. $\endgroup$ Commented Apr 24, 2016 at 2:41

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If $f$ is analytic, so is the real-valued function

$$g(z) = \frac{f(z)}{1 + ik}$$

Now there's a straightforward application of the Cauchy-Riemann equations to conclude that a real-valued analytic function is constant.

Notice that there is one value of $k$ for which this doesn't work.

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