In the book of Flajolet and Sedgewick (this context is not so important, though), the following argumentation is used:
Let $y(z)$ be a function given implicitly by $y - \phi(z,y) = 0$, where $\phi$ is a polynomial. Let $\rho$ be a singularity of $y(z)$ and $\tau = y(\rho)$. At $(\rho,\tau)$, the equation defining $y(z)$ is known to have a multivariate Taylor expansion of the form $$-(z-\rho)\phi'_z(\rho,\tau) - \frac{1}{2}(y-\tau)^2 \phi''_{yy}(\rho,\tau) + \ldots = 0,$$ where $\phi_{yy}''(\rho,\tau)$ is non-zero (and there is no term corresponding to $\phi'_y$). It follows from the context that in general, the three dots may hide even some other terms corresponding to second derivatives.
Then, it somehow should be implied that $y(z)$ has a singular expansion of the form $$y - \tau = -\gamma(1-z/\rho)^{1/2} + \ldots$$ However, I have a trouble understanding why. Could somebody hint a reason behind this? Thank you very much.