We prove that there exists a one-parameter meromorphic solution $u(\tau)$ vanishing at $\tau=0$ 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 formal monodromy parameter $a=\pm i/2$. We study number-theoretic properties of the coefficients of the Taylor-series expansion of $u(\tau)$ at $\tau=0$ and its asymptotic behaviour as $\tau\to+\infty$. These asymptotics are visualized for generic initial data.
Categories: math.CA math-ph math.MP nlin.SI