I want to solve the following equation:
where $f$ and $g$ are defined in the complex plane with real values and they are not analytic.
My question is:
If I assume that $f(s)≠0$, can I deduce that $g(s)=0$ without any further complications?
I am a little confused about this case: if $f=x+iy$ and $g=u+iv$, then $fg=ux-vy+i(uy+vx)$ and $fg$ can be zero if $ux-vy=0,uy+vx=0$ without the implications: $x=y=0,u=v=0$