The discovery of a virtual turning point truly is a breakthrough in WKB analysis of higher
order differential equations. This monograph expounds the core part of its theory together with
its application to the analysis of higher order Painlevé equations of the Noumi-Yamada type and
to the analysis of non-adiabatic transition probability problems in three levels. As M.V.
Fedoryuk once lamented global asymptotic analysis of higher order differential equations had
been thought to be impossible to construct. In 1982 however H.L. Berk W.M. Nevins and K.V.
Roberts published a remarkable paper in the Journal of Mathematical Physics indicating that the
traditional Stokes geometry cannot globally describe the Stokes phenomena of solutions of
higher order equations a new Stokes curve is necessary.
