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]).

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)