Stokes phenomenon

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The property that a function may have different asymptotic expressions when in different domains of the complex -plane. G. Stokes demonstrated [1] that the solution of the so-called Airy equation

which decreases for real , has the following asymptotic expansion when :

where is a constant. The function 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]).


[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)


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].


[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)
How to Cite This Entry:
Stokes phenomenon. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article