I have a question. Given a real function $g(x)$, is it possible to find some constant $a$ and a real function $f$ such that
$$\Im f(a+ix)=g(x)$$ For example,
$$f(x)=e^x,~~~~\Im f(ix)=\Im e^{ix}=\sin x=g(x)$$
and
$$f(x)=\ln(1+x),~~~\Im f(ix)=\arctan x=g(x)$$
It seems this works because $\sin x$ and $\arctan x$ are odd functions, but I can't rigorously prove it.
If $g(x)$ is not an odd function, for example, $g(x)=\ln x$, is it possible to find a constant $a$ and a real function $f$ such that
$$\Im f(a+ix)=\ln x$$