The inverse contour formula recovers a local inverse value

On the circle $\mathrm{\mid }z\mathrm{\mid }=1\mathrm{/}4$ one has $\mathrm{\mid }{z}^2-w\mathrm{\mid }\le \mathrm{\mid }z{\mathrm{\mid }}^2+\mathrm{\mid }w\mathrm{\mid }<\frac{1}{16}+\frac{1}{8}=\frac{3}{16}<\frac{1}{4}=\mathrm{\mid }z\mathrm{\mid }\mathrm{.}$ So Rouché's theorem applied to $z$ and $z+z^2-w$ shows that $f(z)-w$ has exactly one zero inside $\gamma$.

The two roots of $z^2+z-w$ are $(-1\pm s)\mathrm{/}2$. Since $\mathrm{Re}s>0$, one has $\mathrm{\mid }s+1\mathrm{\mid }\ge 1+\mathrm{Re}s>1$, so $\mathrm{\mid }s-1\mathrm{\mid }=\frac{\mathrm{\mid }{s}^2-1\mathrm{\mid }}{\mathrm{\mid }s+1\mathrm{\mid }}=\frac{4\mathrm{\mid }w\mathrm{\mid }}{\mathrm{\mid }s+1\mathrm{\mid }}<\frac{1}{2}\mathrm{.}$ Therefore $\mid \frac{-1+s}{2}\mid <\frac{1}{4},\mid \frac{-1-s}{2}\mid =\frac{\mathrm{\mid }1+s\mathrm{\mid }}{2}>\frac{1}{2}>\frac{1}{4}\mathrm{.}$ So the unique zero inside $\gamma$ is $z(w)=(-1+s)\mathrm{/}2$.

If $f(z)=w$ and $\mathrm{\mid }z\mathrm{\mid }<1\mathrm{/}4$, then $\mathrm{\mid }1+2z\mathrm{\mid }\ge 1-2\mathrm{\mid }z\mathrm{\mid }>1\mathrm{/}2$, so the enclosed zero is simple. The positively oriented circle $\gamma (t)=\frac{1}{4}{e}^{it}$ winds once around every point of $\mathrm{\mid }z\mathrm{\mid }<1\mathrm{/}4$, so the hypotheses of [L1] are satisfied and the inverse contour formula gives $z(w)=\frac{1}{2\pi i}{\int }_{\gamma }\frac{\zeta (1+2\zeta )}{\zeta +{\zeta }^2-w}d\zeta \mathrm{.}$