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{{DEF}}
  
The property that a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090320/s0903201.png" /> may have different asymptotic expressions when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090320/s0903202.png" /> in different domains of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090320/s0903203.png" />-plane. G. Stokes demonstrated [[#References|[1]]] that the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090320/s0903204.png" /> of the so-called [[Airy equation|Airy equation]]
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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) $
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of the so-called [[Airy equation]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090320/s0903205.png" /></td> </tr></table>
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$$
 +
w  ^ {\prime\prime} - zw  = 0
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$$
  
which decreases for real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090320/s0903206.png" />, has the following asymptotic expansion when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090320/s0903207.png" />:
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which decreases for real $  z = x \rightarrow + \infty $,  
 +
has the following asymptotic expansion when $  | z | \rightarrow \infty $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090320/s0903208.png" /></td> </tr></table>
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$$
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w _ {0} ( z)  \sim  Cz  ^ {-1/4}  \mathop{\rm exp} \left ( -
 +
\frac{2}{3}
 +
z  ^ {3/2} \right ) ,
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090320/s0903209.png" /></td> </tr></table>
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$$
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|  \mathop{\rm arg}  z |  \leq  \pi - \epsilon  < \pi ;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090320/s09032010.png" /></td> </tr></table>
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$$
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w _ {0} ( z)  \sim  Ce ^ {i \pi /4 } z  ^ {-1/4} \
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\cos \left (
 +
\frac{2}{3}
 +
z  ^ {3/2} -
 +
\frac \pi {4}
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\right ) ,
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090320/s09032011.png" /></td> </tr></table>
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$$
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|  \mathop{\rm arg}  z - \pi |  \leq  \epsilon  < \pi ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090320/s09032012.png" /> is a constant. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090320/s09032013.png" /> is an entire function, while its asymptotic expansion is a discontinuous function.
+
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]]]).
 
The Stokes phenomenon also occurs for Laplace integrals, solutions of ordinary differential equations, etc. (see [[#References|[2]]], [[#References|[3]]]).
 
====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></table>
 
 
 
  
 
====Comments====
 
====Comments====
Line 28: Line 59:
  
 
====References====
 
====References====
<table><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>
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<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>

Latest revision as of 09:42, 14 April 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=26717
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article