Positive function for complex variable

Question

I was curious whether it makes sense to talk about positive functions of a complex variable. I.e, does it make sense to talk about $f(z) > 0$, if $z$ is complex, say? As an example, what if $f(z) = \cos z + i \sin z$?

Answer 1

Only if $f(z)$ is a real number. If it's a complex number, then no. See discussion here.