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If f(z) is an entire function, which gets only real values for real z, and

$$ f(0)=0,$$ $$f'(0)\ne 0$$

and the Image of the imagainary axie is a straight line, then this line is the imagainary axie.

True or False?

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Please consider changing the title to something more specific. –  beauby Nov 24 '12 at 20:15

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True.

Since $f'(0) \in \mathbb R$, you know that $\lim\limits_{h \rightarrow 0}\frac{f(ih)}{h}$ is imaginary, so the only straight line in question is the imaginary axis.

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