Does there exists an entire function $f: \mathbb C \to \mathbb C$ which is bounded on real line and imaginary line?
Does there exists a nonconstant entire function $f: \mathbb C \to \mathbb C$ which is bounded on real line and imaginary line?
Clearly,$ f(z)=sin(z)$ is an example of an entire function which is bounded on real line and $ f(z)= e^z$ is example of a function which is bounded on imaginary line.But I'm unable to find a function which is bounded on both the lines.Any ideas?
Solution 1:
$f(z) =e^{iz^2}$ will do that for you.