line of given imaginary part; these choices differ only by a vertical shift, showing that there is no canonical choice for the Stokes line. Example: the Airy function The Airy function Ai(x) 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 (when x crosses a branch cut) because these approximations contain multi-valued functions. In contrast, the Airy function is single valued and indeed entire and therefore, in order to make sense of the approximation, one has to choose a single value out of the multiple possible values (this imposes a branch cut for the approximation, by implication). 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 (rotating it around through the upper half complex plane) 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 because 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. Example: second order linear differential equations 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. (Choosing different values of a is equivalent to choosing different coefficients in the linear combination.) The anti-Stokes lines and Stokes lines are then the zeros of the real