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Difference between revisions of "Stokes phenomenon"

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in different domains of the complex  $  z $-
 
in different domains of the complex  $  z $-
 
plane. G. Stokes demonstrated [[#References|[1]]] that the solution  $  w _ {0} ( z) $
 
plane. G. Stokes demonstrated [[#References|[1]]] that the solution  $  w _ {0} ( z) $
of the so-called [[Airy equation|Airy equation]]
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of the so-called [[Airy equation]]
  
 
$$  
 
$$  
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====References====
 
====References====
 
<table>
 
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  G.G. Stokes, ''Trans. Cambridge Philos. Soc.'' , '''10''' (1864)  pp. 106–128</TD></TR>
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<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">[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">[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>
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<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>
+
<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>
 
</table>

Revision as of 06:48, 24 February 2024


This page is deficient and requires revision. Please see Talk:Stokes phenomenon for further comments.

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)
How to Cite This Entry:
Stokes phenomenon. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stokes_phenomenon&oldid=55555
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article