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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 [[#References|[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 [[#References|[2]]], [[#References|[3]]]).

====Comments====
There is a recent interest in the Stokes phenomenon in asymptotic analysis, which is initiated by M.V. Berry in [[#References|[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 [[#References|[a2]]].

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