Let f(z) be a holomorphic function which takes only pure imaginary values. Show that f is constant.
I am a little stuck with this as I have noted that $z=u+iv$ where $u=0$ hence $du/dx=du/dy=0$ so by the Cauchy riemann equations $dv/dx=dv/dy=0$ and hence integration we have v is a constant but I'm lost on how to show that f is constant?
where do we take the function in this?