Suppose that ff is analytic on some region R∈CR\in\mathbb{C}. If Im(f)(f) = k⋅k\cdotRe(f)(f) for some nonzero constant k∈Ck\in\mathbb{C}, then show that ff is constant on RR.

I know that if f′(z)=0f'(z)=0 for all zz in some region RR, then f(z)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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The constant kk had better be a real number, since both Im(f)\text{Im}(f) and Re(f)\text{Re}(f) are real-valued.

– Ted Shifrin

Apr 24 at 2:41

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1 Answer

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If ff is analytic, so is the real-valued function

g(z)=f(z)1+ikg(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 kk for which this doesn’t work.