Department of Applied Mathematics
University of Colorado
Boulder, Colorado, USA
The six Painlevé transcendents P_I to P_VI have both applications and
analytic properties that make them stand out from most other classes of
special functions. Although they have been the subject of extensive
theoretical investigations for about a century, they still have a
reputation of being numerically challenging. In particular, their
extensive pole fields in the complex plane have often been perceived as
'numerical mine fields'. We note in this present work that, on the
contrary, the Painlevé property in fact provides excellent opportunities
for very fast and accurate numerical solutions across the full complex
plane. The numerical method will be illustrated for the P_I equation.
The present work was carried out in collaboration with Prof. André Weideman (University of Stellenbosch). Jonah Reeger (CU Boulder) has assisted vith creating movie clips.