Somewhat confusingly, mathematicians and physicists use the terms "Stokes line" and "anti-Stokes line" in opposite ways. The lines originally studied by Stokes are what some mathematicians call anti-Stokes lines and what physicists call Stokes lines.. This article uses the physicist's convention, which is historically more accurate and seems to be becoming more common among mathematicians. recommends the term "principal curve" for anti-Stokes lines. Informally the anti-Stokes lines are roughly where some term in the asymptotic expansion changes from increasing to decreasing, and the Stokes lines are lines along which some term approaches infinity or zero fastest. Anti-Stokes lines bound regions where the function exhibit a particular asymptotic behavior. The Stokes lines and anti-Stokes lines are not unique and do not really have a precise definition in general, because the region where a function has a given asymptotic behavior is a somewhat vague concept. However the lines do usually have well determined directions at essential singularities of the function, and there is sometimes a natural choice of these lines as follows. The asymptotic expansion of a function is often given by a linear combination of functions of the form fe±g for functions f and g. The Stokes lines can then be taken as the zeros of the imaginary part of g, and the anti-Stokes lines as the zeros of the real part of g. If the lines are defined like this then they are orthogonal where they meet, unless g has a multiple zero. As a trivial example, the function sinh has two regions Re > 0 and Re < 0 where it is asymptotic to ez/2 and −e−z/2. So the anti-Stokes line can be taken to be the imaginary axis, and the Stokes line can be taken to be the real axis. One could equally well take the Stokes line to be any line of given imaginary part; these choices differ only by a vertical shift, showing that there is no canonical choice for the Stokes line.
The Airy function Ai is one of two solutions to a simple differential equation which it is often useful to approximate for many values of x - including complex values. For large x of given argument the solution can be approximated by a linear combination of the functions However the linear combination has to change as the argument of x passes certain values because these approximations are multi-valued functions but the Airy function is single valued. For example, if we regard the limit of x as large and real, and would like to approximate the Airy function for both positive and negative values, we would find that which are two very different expressions. What has happened is that as we have increased the argument of x from 0 to pi we have crossed an anti-Stokes line, which in this case is at. At this anti-Stokes line, the coefficient of is forced to jump. The coefficient of can jump at this line but is not forced to; it can change gradually as arg x varies from π/3 to π as it is not determined in this region. There are three anti-Stokes lines with arguments π/3, π. –π/3, and three Stokes lines with arguments 2π/3, 0. –2π/3.
The Airy function example can be generalized to a broad class of second order linear differential equations as follows. By standard changes of variables, a second order equation can often be changed to one of the form where f is holomorphic in a simply-connected region and w is a solution of the differential equation. Then in some cases the WKB method gives an asymptotic approximation for w as a linear combination of functions of the form for some constant a. The anti-Stokes lines and Stokes lines are then the zeros of the real and imaginary parts, respectively, of If a is a simple zero of f then locally flooks like. Solutions will locally behave like the Airy functions; they will have three Stokes lines and three anti-Stokes lines meeting at a.
