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?
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?
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.