Various properties of algebroid solutions of the degenerate third Painlev\'e equation, \begin{ equation*} u^{ \prime \prime}(\tau) \! = \! \frac{ (u^{ \prime}(\tau))^{ 2}}{ u(\tau)} \! - \! \frac{ u^{ \prime}(\tau)}{ \tau} \! + \! \frac{ 1}{ \tau} \! \left(-8 \varepsilon (u(\tau))^{ 2} \! + \! 2ab \right) \! + \! \frac{ b^{ 2}}{ u(\tau)},\qquad \varepsilon=\pm1,\quad\varepsilon b>0, \end{ equation*} for the monodromy parameter $a=0$ are studied. The paper contains connection results for asymptotics as $\tau\to+0$ and as $\tau\to+\infty$ for $a\in\mathbb{ C}$. Using these results, the simplest algebroid solution with asymptotics $u(\tau)\to c\tau^{ 1/3}$ as $\tau\to0$, where $c\in\mathbb{ C}\setminus\{ 0\}$, together with its associated integral $\int_0^\tau { (u(t))^{ -1}\,d t}$, are considered in detail, and their basic asymptotic behaviours are visualized.
Categories: math.CA math-ph math.MP