The Degenerate Third Painleve' Equation: Complete Asymptotic Classification of Solutions in the Neighbourhood of the Regular Singular Point

We give a classification for the small-$τ$ asymptotic behaviours of solutions to the degenerate third Painlevé equation, $u^{''}(τ) = \frac{(u^{\prime}(τ))^{2}}{u(τ)} - \frac{u^{\prime}(τ)}τ + \frac{1}τ\left(-8 \varepsilon (u(τ))^{2} + 2ab \right) + \frac{b^{2}}{u(τ)}, \quad\varepsilon=\pm1,\quad\varepsilon b>0, \quad a\in\mathbb{C}\setminus i\mathbb{Z}$, in terms of the monodromy data of a $2\times2$ matrix linear ODE whose isomonodromy deformations they describe. We also study the complete asymptotic expansions of the solutions.