A complex equation I want to solve the following equation:
$g(s)f(s)=0$
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: $x=y=0,u=v=0$

The point is that the complex numbers form a field.  Doing it with  real and imaginary parts is the hard way.  Simpler: note that $z^{-1} = \overline{z}/|z|^2$ is the multiplicative inverse for $z \ne 0$, since $z \overline{z} = |z|^2$ (here if $z = x + i y$, $\overline{z} = x - i y$ and $|z| = \sqrt{x^2 + y^2}$).  So if $wz = 0$ and $z \ne 0$, multiplying both sides by $z^{-1}$ we get $w = 0$.
