Stokes phenomenon

In complex analysis the Stokes phenomenon, discovered by G. G. Stokes (1847, 1858), is that the asymptotic behavior of functions can differ in different regions of the complex plane. These regions are bounded by Stokes line or anti-Stokes lines.

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. (These terms are also used in optics for the unrelated Stokes lines and anti-Stokes lines in Raman scattering). This article uses the physicist's convention, which is historically more accurate and seems to be becoming more common among mathematicians. Olver (1997) recommends the term "principal curve" for (physicist's) 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 has some 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 f(z)e^±g(z) 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. (This is not quite canonical, because one can add a constant to g, changing the lines.) 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(z) has two regions Re(z) > 0 and Re(z) < 0 where it is asymptotic to e^z/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 change of variables, showing that there is no canonical choice for the Stokes line.

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