The concept of Stokes line width is introduced for the asymptotic expa
nsions of functions near an essential singularity. Explicit expression
s 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 fun
ction, not associated with expansion in an asymptotic series, in the f
orm of dominant and recessive terms is obtained by a special division
of the contour integral which represents the function into contributio
ns of higher and lower saddle points. This division leads to a narrowe
r, natural Stokes line width and a switching function of an argument t
hat depends on the topology of the lines of steepest descent from the
saddle point.