Stokes phenomenon

The Stokes phenomenon is a change in the asymptotic contribution of a subdominant exponential function as a complex parameter crosses a distinguished direction. A function can remain holomorphic across that direction even though different sectorial asymptotic expansions describe it most effectively. An inverse-power expansion alone cannot determine contributions smaller than every inverse power.

A concrete mechanism appears in the Gaussian Cauchy transform and its Stokes jump. Continuing a contour past a simple pole changes the represented contour integral by $2\pi i$ times its residue; here that term is proportional to $e^{-z^2}$. In directions where $\mathrm{Re}(z^2)>0$, this exponential is invisible to every algebraic order. The Borel integral with a simple pole gives the same mechanism in the transformed plane.

A Stokes line uses the equal-phase convention adopted here, while an anti-Stokes line marks equal exponential magnitudes. Those names are interchanged in some sources, so the actions or defining equations should accompany them. Near a switching direction, optimal truncation and error-function smoothing of a Stokes multiplier give a continuous transition instead of a literal jump of the underlying analytic function.^[ref]