It is impossible for the smooth rolling-up spiral to be a solution of the Birkhoff-Rott equation, because of the finite-time singularity. However, there is a possibility to analytically continue the Birkhoff-Rott equation to the spiral along the path in complex time to get around the singularity.
The purpose of the study is to discuss wheter the analytic continuation is possible or not. We investigate numerically the critical times in the complex-time plane at which the solution of the regularized Birkhoff-Rott equation of Krasny's type. The study indicates that it is impossible to analytically continue the Birkhoff-Rott equation to the smooth rolling-up spiral along any path in complex time.
In addition, we study a simple model to approximate the rolling-up vortex sheet motion. The model has a significant resemblance between the vortex sheet and the model. Comparison between the vortex sheet and the model suggests that the rolling-up spiral with inifinite windings at the center of the spiral could be a solution of the Birkhoff-Rott equation beyond the real singularity time.