 | This article needs attention from an expert in Mathematics. The specific problem is: The only thing I can understand here is that "something makes the lines curvy..." further to this both examples on this page are extremely erroneous. (June 2023) |
In complex analysis the Stokes phenomenon, discovered by G. G. Stokes (1847, 1858), is where the asymptotic behavior of functions can differ in different regions of the complex plane. This seemingly gives rise to a paradox when looking at the asymptotic expansion of an analytic function. Since an analytic function is continuous you would expect the asymptotic expansion to be continuous. This paradox is the subject of Stokes' early research and is known as Stokes phenomenon. The regions in the complex plane with different asymptotic behaviour are bounded by possibly one or two types of curves known as Stokes curves and Anti-Stokes Curves. This apparent paradox has since been resolved and the supposed discontinuous jump in the asymptotic expansions has been shown to be smooth and continuous. In order to resolve this paradox the asymptotic expansion needs to be handled in a careful manner. More specifically the asymptotic expansion must include additional exponentially small terms relative to the usual algebraic terms included in a usual asymptotic expansion. What happens in Stokes phenomenon is that an asymptotic expansion in one region may contain an exponentially small contribution (neglecting this contribution still gives a correct asymptotic expansion for that region). However, this exponentially small term can become exponentially large in another region of the complex plane, this change occurs across the Anti-Stokes curves. Furthermore the exponentially small term may switch on or off other exponentially small terms, this change occurs across a Stokes curve. Including these exponentially small terms allows the asymptotic expansion to be written as a continuous expansion for the entire complex domain which resolves the Stokes Phenomenon paradox.