Stokes line width

The concept of Stokes line width is introduced for the asymptotic expansions of functions near an essential singularity. Explicit expressions are found for functions (switching functions) that “switch on” the exponentially small terms for the Dawson integral, Airy function, and the gamma function. A different, more natural representation of a function, not associated with expansion in an asymptotic series, in the form of dominant and recessive terms is obtained by a special division of the contour integral which represents the function into contributions of higher and lower saddle points. This division leads to a narrower, natural Stokes line width and a switching function of an argument that depends on the topology of the lines of steepest descent from the saddle point.