Stokes phenomenon

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The property that a function $ f( z) $ may have different asymptotic expressions when $ | z | \rightarrow \infty $ in different domains of the complex $ z $- plane. G. Stokes demonstrated [1] that the solution $ w _ {0} ( z) $ of the so-called Airy equation

$$ w ^ {\prime\prime} - zw = 0 $$

which decreases for real $ z = x \rightarrow + \infty $, has the following asymptotic expansion when $ | z | \rightarrow \infty $:

$$ w _ {0} ( z) \sim Cz ^ {-1/4} \mathop{\rm exp} \left ( - \frac{2}{3} z ^ {3/2} \right ) , $$

$$ | \mathop{\rm arg} z | \leq \pi - \epsilon < \pi ; $$

$$ w _ {0} ( z) \sim Ce ^ {i \pi /4 } z ^ {-1/4} \ \cos \left ( \frac{2}{3} z ^ {3/2} - \frac \pi {4} \right ) , $$

$$ | \mathop{\rm arg} z - \pi | \leq \epsilon < \pi , $$

where $ C \neq 0 $ is a constant. The function $ w _ {0} ( z) $ is an entire function, while its asymptotic expansion is a discontinuous function.

The Stokes phenomenon also occurs for Laplace integrals, solutions of ordinary differential equations, etc. (see [2], [3]).

Comments

There is a recent interest in the Stokes phenomenon in asymptotic analysis, which is initiated by M.V. Berry in [a1]. In the new interpretation of the phenomenon, an error function is introduced to describe the rapid change in the behaviour of the remainders of the asymptotic expansions as a Stokes line is crossed. A rigorous treatment of Berry's observation is given in [a2].

References

[1]	G.G. Stokes, Trans. Cambridge Philos. Soc. , 10 (1864) pp. 106–128
[2]	J. Heading, "An introduction to phase-integral methods" , Methuen (1962)
[3]	N.G. de Bruijn, "Asymptotic methods in analysis" , Dover, reprint (1981)
[a1]	M.V. Berry, "Uniform asymptotic smoothing of Stokes' discontinuities" Proc. R. Soc. London A , 422 (1989) pp. 7–21
[a2]	"On Stokes's phenomenon and converging factors" R. Wong (ed.), Proc. Int. Symp. Asymptotic and Computational Anal. (Winnipeg, Manitoba) , M. Dekker (1990)