A question about the Residue of $h=fg$ Let $f$ and $g$ be two functions (not necessarily analytic) of the complex
variable $z$ such that for some $\varepsilon >0$ :
1) $f$ is continuous on $0<\left\vert z\right\vert <\varepsilon $
2) $g$ is continuous on $\left\vert z\right\vert <\varepsilon $ and $%
g\left( 0\right) \neq 0$
3) $h=fg$ is analytic on $0<\left\vert z\right\vert <\varepsilon .$
Do we have 
\begin{equation*}
Res\left( h,0\right) =\frac{g\left( 0\right) }{2\pi i}\oint_{\left\vert
z\right\vert =\frac{\varepsilon }{2}}f\left( z\right) dz
\end{equation*}
Or perhaps some other equation ?
Thank you !
 A: The answer to the question as you stated is no. This is why I doubt it is true. If it were true, then it must be that for all $\eta<\epsilon$ $$\text{Res}\left( h,0\right) =\frac{g\left( 0\right) }{2\pi i}\oint_{\left\vert
z\right\vert =\frac{\eta}{2}}f\left( z\right) dz.$$ The right-hand side may depend on $\eta$ if $f$ is not analytic but the left-hand side remains constant. So I guess more meaningful (and weaker) version of your question is to ask whether
$$
\text{Res}\left( h,0\right) =\frac{g\left( 0\right) }{2\pi i}\lim_{\eta \to 0}\oint_{\left\vert
z\right\vert =\eta}f\left( z\right) dz
$$ holds. Since $\text{Res}\left( h,0\right) =\frac{1}{2\pi i}\oint_{\left\vert
z\right\vert =\eta}f\left(z\right)g(z) dz$ for all $\eta<\epsilon$, it is asking whether we can treat $g(z)$ as if it is constant $g(0)$ as $\eta \to 0$. But it turns out that it is also false even if we assume $f$ and $g$ are analytic.  A counterexample is $g(z) = \frac{1}{1+z}$ and $h(z) = \frac{1}{z}+\frac{1}{z^2}$. Then $\text{Res}\left( h,0\right) = 1$ while
$$
\frac{g(0)}{2\pi i}\lim_{\eta \to 0}\oint_{\left\vert
z\right\vert =\eta}f\left( z\right) dz = \frac{g(0)}{2\pi i}\lim_{\eta \to 0}\oint_{\left\vert
z\right\vert =\eta}(1+z)(\frac{1}{z}+\frac{1}{z^2}) dz =2.
$$
Note: However, if we assume that $h$ has a simple pole at $z=0$ and $\frac{1}{g(z)} = \frac{1}{g(0)}+\mathcal{O}(|z|^{\epsilon'})$ for some $\epsilon'>0$, then $\text{Res}\left( h,0\right) =\frac{g\left( 0\right) }{2\pi i}\lim_{\eta \to 0}\oint_{\left\vert
z\right\vert =\eta}f\left( z\right) dz$ must be true.
