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Particular solutions of the [[Airy equation|Airy equation]].
 
Particular solutions of the [[Airy equation|Airy equation]].
  
 
The first Airy function (or simply the Airy function) is defined by
 
The first Airy function (or simply the Airy function) is defined by
  
<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/a/a011/a011210/a0112101.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Ai} (x)  =
 +
\frac{1} \pi
 +
\int\limits _ { 0 } ^  \infty 
 +
\cos \left (
 +
\frac{t  ^ {3} }{3}
 +
+ x t \right ) \
 +
d t .
 +
$$
  
For complex values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a0112102.png" />
+
For complex values of $  z $
  
<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/a/a011/a011210/a0112103.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Ai} (z)  =
 +
\frac{1}{2 \pi }
 +
\int\limits _  \gamma
 +
\mathop{\rm exp} \left ( zt -  
 +
\frac{t  ^ {3} }{3}
 +
\right ) \
 +
d t ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a0112104.png" /> is a contour in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a0112105.png" />-plane. The second Airy function is defined by
+
where $  \gamma = ( \infty e ^ {- 2 \pi i / 3 } , 0 ] \cup [ 0 , + \infty ) $
 +
is a contour in the complex $  t $-
 +
plane. The second Airy function is defined by
  
<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/a/a011/a011210/a0112106.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Bi} (z)  = i \omega  ^ {2}  \mathop{\rm Ai} ( \omega  ^ {2} z ) -
 +
i \omega  \mathop{\rm Ai} ( \omega z ) ,\  \omega  = e ^ {2 \pi i / 3 } .
 +
$$
  
The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a0112107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a0112108.png" /> are real for real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a0112109.png" />.
+
The functions $  \mathop{\rm Ai} (x) $
 +
and $  \mathop{\rm Bi} (x) $
 +
are real for real $  x $.
  
 
A second collection of Airy functions was introduced by V.A. Fock [V.A. Fok]:
 
A second collection of Airy functions was introduced by V.A. Fock [V.A. Fok]:
  
<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/a/a011/a011210/a01121010.png" /></td> </tr></table>
+
$$
 +
v (z)  =
 +
\frac{\sqrt \pi }{2}
 +
  \mathop{\rm Ai} (z) ,
 +
$$
 +
 
 +
$$
 +
w _ {1} (z)  = 2 e ^ {i \pi / 6 } v ( \omega z ) ,
 +
$$
  
<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/a/a011/a011210/a01121011.png" /></td> </tr></table>
+
$$
 +
w _ {2} (z)  = 2 e ^ {- i \pi / 6 } v ( \omega  ^ {-1} z ) ;
 +
$$
  
<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/a/a011/a011210/a01121012.png" /></td> </tr></table>
+
in this case  $  v (z) $
 +
is called the Airy–Fok function (Airy–Fock function). The following identities hold:
  
in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121013.png" /> is called the Airy–Fok function (Airy–Fock function). The following identities hold:
+
$$ \tag{1 }
 +
v (z)  =
 +
\frac{w _ {1} (z) - w _ {2} (z) }{2i}
 +
,\ \
 +
\overline{ {w _ {1} (z) }}\;  = w _ {2} ( \overline{z}\; ) .
 +
$$
  
<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/a/a011/a011210/a01121014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
Any two of  $  v (z) , w _ {1} (z) $
 +
and  $  w _ {2} (z) $
 +
are linearly independent.
  
Any two of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121016.png" /> are linearly independent.
+
The most important Airy function is  $  v (z) $(
 +
or  $  \mathop{\rm Ai} (z) $).  
 +
Its asymptotic behaviour on the real axis is given by
  
The most important Airy function is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121017.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121018.png" />). Its asymptotic behaviour on the real axis is given by
+
$$
 +
v (x)  =
 +
\frac{1}{2}
 +
 +
\frac{x  ^ {-1/4} }{\sqrt \pi }
 +
\
 +
\mathop{\rm exp} \left ( -
 +
\frac{2}{3}
 +
x  ^ {3/2} \right ) [ 1 +
 +
O ( x  ^ {-3/2} ) ] ,
 +
\  x \rightarrow + \infty ,
 +
$$
  
<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/a/a011/a011210/a01121019.png" /></td> </tr></table>
+
$$
 +
v (x)  =
 +
\frac{| x |  ^ {-1/4} }{\sqrt \pi }
 +
\left [
 +
\sin \left (
 +
\frac{2}{3}
 +
| x |  ^ {3/2} +
 +
\frac \pi {4}
  
<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/a/a011/a011210/a01121020.png" /></td> </tr></table>
+
\right ) + O ( | x |  ^ {-3/2} ) \right ] ,
 +
$$
  
<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/a/a011/a011210/a01121021.png" /></td> </tr></table>
+
$$
 +
\  x \rightarrow - \infty ,
 +
$$
  
so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121022.png" /> decreases rapidly for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121023.png" /> and oscillates strongly for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121024.png" />. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121026.png" /> increase exponentially as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121027.png" />. For complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121028.png" /> the Airy functions have the following asymptotic expansions as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121029.png" />:
+
so $  v (x) $
 +
decreases rapidly for $  x > 0 , x \gg 1 $
 +
and oscillates strongly for $  x < 0 , | x | \gg 1 $.  
 +
The functions $  w _ {1} (x) $
 +
and $  w _ {2} (x) $
 +
increase exponentially as $  x \rightarrow + \infty $.  
 +
For complex $  z $
 +
the Airy functions have the following asymptotic expansions as $  | 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/a/a011/a011210/a01121030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
v (z)  \sim 
 +
\frac{1}{2 \sqrt \pi }
 +
z  ^ {-1/4}  \mathop{\rm exp}
 +
\left ( -
 +
\frac{2}{3}
 +
z  ^ {3/2} \right ) \times
 +
$$
  
<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/a/a011/a011210/a01121031.png" /></td> </tr></table>
+
$$
 +
\times
 +
\sum _ { n=0 } ^  \infty  (-1)  ^ {n} a _ {n} z  ^ {-3n/2}
 +
\  \textrm{ for }  |  \mathop{\rm arg}  z | \leq  \pi - \epsilon ,
 +
$$
  
<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/a/a011/a011210/a01121032.png" /></td> </tr></table>
+
$$
 +
w _ {1} (z)  \sim 
 +
\frac{1}{\sqrt \pi }
 +
z  ^ {-1/4}
 +
  \mathop{\rm exp} \left (
 +
\frac{2}{3}
 +
z  ^ {3/2} \right ) \times
 +
$$
  
<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/a/a011/a011210/a01121033.png" /></td> </tr></table>
+
$$
 +
\times
 +
\sum _ { n=0 } ^  \infty  a _ {n} z  ^ {-3n/2} \  \textrm{ for } \
 +
|  \mathop{\rm arg}  z -
 +
\frac \pi {3}
 +
| \leq  \pi - \epsilon ,
 +
$$
  
 
where
 
where
  
<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/a/a011/a011210/a01121034.png" /></td> </tr></table>
+
$$
 +
a _ {n}  = \
 +
 
 +
\frac{\Gamma \left ( 3 n +
 +
\frac{1}{2}
 +
\right ) 9  ^ {-n} }{( 2 n ) ! }
 +
.
 +
$$
  
The asymptotic expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121035.png" /> is of the form (2), but it is valid in the sector
+
The asymptotic expansion of $  w _ {2} (z) $
 +
is of the form (2), but it is valid in the sector
  
<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/a/a011/a011210/a01121036.png" /></td> </tr></table>
+
$$
 +
\left |  \mathop{\rm arg}  \left ( z +
 +
\frac \pi {3}
 +
\right ) \
 +
\right |  \leq  \pi - \epsilon .
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121037.png" /> is arbitrary, the branches of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121039.png" /> are positive on the semi-axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121040.png" />, and the asymptotic expansions are uniform with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121041.png" /> and can be differentiated term by term any number of times. In the remaining sector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121042.png" /> the asymptotic expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121043.png" /> is expressed in terms of those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121045.png" /> by means of (1); hence, the asymptotic expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121046.png" /> has a different form in different sectors of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121047.png" />-plane. This fact was first established by G.G. Stokes [[#References|[2]]] and is called the Stokes phenomenon.
+
Here $  \epsilon \in ( 0 , \pi ) $
 +
is arbitrary, the branches of $  \sqrt z $
 +
and $  z  ^ {1/4} $
 +
are positive on the semi-axis $  ( 0 , \infty ) $,
 +
and the asymptotic expansions are uniform with respect to $  \mathop{\rm arg}  z $
 +
and can be differentiated term by term any number of times. In the remaining sector $  |  \mathop{\rm arg}  -z | < \epsilon $
 +
the asymptotic expansion of $  v (z) $
 +
is expressed in terms of those of $  w _ {1} (z) $
 +
and $  w _ {2} (z) $
 +
by means of (1); hence, the asymptotic expansion of $  v (z) $
 +
has a different form in different sectors of the complex $  z $-
 +
plane. This fact was first established by G.G. Stokes [[#References|[2]]] and is called the Stokes phenomenon.
  
 
The Airy functions occur in the study of integrals of rapidly-oscillating functions, of the form
 
The Airy functions occur in the study of integrals of rapidly-oscillating functions, of the form
  
<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/a/a011/a011210/a01121048.png" /></td> </tr></table>
+
$$
 +
I ( \lambda , \alpha )  = \int\limits _ { a } ^ { b }
 +
\textrm{ e } ^ {i \lambda S ( x , \alpha ) }
 +
f ( x , \alpha )  d x ,
 +
$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121049.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121051.png" /> are smooth functions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121052.png" /> is real and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121053.png" /> is a real parameter. If for small values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121054.png" /> the phase <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121055.png" /> has two close non-degenerate stationary points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121057.png" /> that coincide for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121058.png" />, for example, if
+
for $  \lambda > 0 , \lambda \rightarrow \infty $.  
 +
Here $  f $
 +
and $  S $
 +
are smooth functions, $  S $
 +
is real and $  \alpha $
 +
is a real parameter. If for small values of $  \alpha \geq  0 $
 +
the phase $  S $
 +
has two close non-degenerate stationary points $  x _ {1} ( \alpha ) $
 +
and $  x _ {2} ( \alpha ) $
 +
that coincide for $  \alpha = 0 $,  
 +
for example, if
  
<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/a/a011/a011210/a01121059.png" /></td> </tr></table>
+
$$
 +
S ( x , \alpha )  = \alpha x - x  ^ {3} + O
 +
( x  ^ {4} ) \  \textrm{ as }  x \rightarrow 0 ,
 +
$$
  
then for small values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121060.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121061.png" />, the contribution to the asymptotics of the integral coming from a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121062.png" /> can be expressed in terms of the Airy function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121063.png" /> and its derivative (see [[#References|[6]]]). Integrals of this kind occur in the study of short-wave fields near a simple focus (see [[#References|[7]]] and [[#References|[8]]]); the Airy functions arose in connection with the study of this problem [[#References|[1]]].
+
then for small values of $  \alpha \geq  0 $,  
 +
as $  \lambda \rightarrow + \infty $,  
 +
the contribution to the asymptotics of the integral coming from a neighbourhood of the point $  x = 0 $
 +
can be expressed in terms of the Airy function $  v $
 +
and its derivative (see [[#References|[6]]]). Integrals of this kind occur in the study of short-wave fields near a simple focus (see [[#References|[7]]] and [[#References|[8]]]); the Airy functions arose in connection with the study of this problem [[#References|[1]]].
  
 
Consider the second-order differential equation
 
Consider the second-order differential 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/a/a011/a011210/a01121064.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
y  ^ {\prime\prime} + \lambda  ^ {2} q (x) y  = 0 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121065.png" /> is a smooth real-valued function on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121067.png" /> is a large parameter. The zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121068.png" /> are called turning points (or transfer points) of the equation (3). Let
+
where $  q (x) $
 +
is a smooth real-valued function on the interval $  I = [ a , b ] $
 +
and  $  \lambda > 0 $
 +
is a large parameter. The zeros of $  q (x) $
 +
are called turning points (or transfer points) of the equation (3). Let
  
<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/a/a011/a011210/a01121069.png" /></td> </tr></table>
+
$$
 +
< x _ {0}  < b ,\ \
 +
q ( x _ {0} )  = 0 ,\ \
 +
q  ^  \prime  ( x _ {0} )  \neq  0
 +
$$
  
 
(such a point is called simple),
 
(such a point is called simple),
  
<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/a/a011/a011210/a01121070.png" /></td> </tr></table>
+
$$
 +
q (x)  \neq  0 \  \textrm{ for } \
 +
x \in I , x \neq x _ {0} ,\ \
 +
q  ^  \prime  ( x _ {0} )  > 0 .
 +
$$
  
 
Set
 
Set
  
<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/a/a011/a011210/a01121071.png" /></td> </tr></table>
+
$$
 +
\xi (x)  = \left (
 +
\frac{2}{3}
 +
\int\limits _ {x _ {0} } ^ { x }
 +
\sqrt {q (t) }  d t \right )  ^ {2/3} ,\ \
 +
\mathop{\rm sign}  \xi (x)  =   \mathop{\rm sign} ( x - x _ {0} ) ,
 +
$$
  
<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/a/a011/a011210/a01121072.png" /></td> </tr></table>
+
$$
 +
Y _ {0} (x)  = ( \xi  ^  \prime  (x) )  ^ {-1/2}  \mathop{\rm Ai} ( - \lambda  ^ {2/3} \xi (x) ) ,
 +
$$
  
<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/a/a011/a011210/a01121073.png" /></td> </tr></table>
+
$$
 +
Y _ {1} (x)  = ( \xi  ^  \prime  (x) )  ^ {-1/2}  \mathop{\rm Bi} ( - \lambda  ^ {2/3} \xi (x) ) .
 +
$$
  
Equation (3) has linearly independent solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121075.png" /> such that, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121076.png" />,
+
Equation (3) has linearly independent solutions $  y _ {0} (x) $
 +
and $  y _ {1} (x) $
 +
such that, as $  \lambda \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/a/a011/a011210/a01121077.png" /></td> </tr></table>
+
$$
 +
y _ {j} (x)  = Y _ {j} (x) \left [ 1 + O \left (
 +
\frac{1} \lambda
 +
\right ) \right ] ,\ \
 +
a \leq  x \leq  x _ {0} ,\  j = 0 , 1 ,
 +
$$
  
<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/a/a011/a011210/a01121078.png" /></td> </tr></table>
+
$$
 +
y _ {0} (x)  = Y _ {0} (x) \left [ 1 + O \left (
 +
\frac{1} \lambda
  
<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/a/a011/a011210/a01121079.png" /></td> </tr></table>
+
\right ) \right ] + Y _ {1} (x) O \left (
 +
\frac{1} \lambda
 +
\right ) ,
 +
$$
  
<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/a/a011/a011210/a01121080.png" /></td> </tr></table>
+
$$
 +
y _ {1} (x)  = Y _ {1} (x) \left [ 1 + O \left (
 +
\frac{1} \lambda
  
uniformly with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121081.png" />.
+
\right ) \right ] + Y _ {0} (x) O \left (
 +
\frac{1} \lambda
 +
\right ) ,
 +
$$
  
This result has been generalized in various directions: asymptotic series have been obtained for the solutions, the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121082.png" /> has been studied (for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121083.png" /> can be expanded in an asymptotic series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121084.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121085.png" />), and the asymptotic behaviour of the solutions near multiple turning points has been investigated. Other generalizations concern the equation
+
$$
 +
x _ {0}  \leq  x  \leq  b ,
 +
$$
  
<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/a/a011/a011210/a01121086.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
uniformly with respect to  $  x $.
  
where the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121087.png" /> is analytic in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121088.png" /> of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121089.png" />-plane. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121090.png" /> be the maximal connected component of the level line
+
This result has been generalized in various directions: asymptotic series have been obtained for the solutions, the case  $  q = q ( x , \lambda ) $
 +
has been studied (for example, if  $  q ( x , \lambda ) $
 +
can be expanded in an asymptotic series  $  q \sim \sum _ {n=0}  ^  \infty  \lambda  ^ {-n} q _ {n} (x) $
 +
as  $  \lambda \rightarrow + \infty $),
 +
and the asymptotic behaviour of the solutions near multiple turning points has been investigated. Other generalizations concern the 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/a/a011/a011210/a01121091.png" /></td> </tr></table>
+
$$ \tag{4 }
 +
w  ^ {\prime\prime} + \lambda  ^ {2} q (x) w  = 0 ,
 +
$$
  
emanating from a turning point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121092.png" /> and containing no other turning points; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121093.png" /> is called a Stokes line. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121094.png" /> (that is, (4) is the Airy equation), then the Stokes lines are the rays <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121096.png" />. Analogously, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121097.png" /> is a simple turning point of (4), then there are three Stokes lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a01121099.png" /> emanating from it and the angle between adjacent lines at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a011210100.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a011210101.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a011210102.png" /> be a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a011210103.png" /> from which a neighbourhood of the Stokes line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a011210104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a011210105.png" />, has been removed. For a suitable numbering of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a011210106.png" />, equation (4) has three solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a011210107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a011210108.png" />, such that, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a011210109.png" />,
+
where the function  $  q (z) $
 +
is analytic in a domain  $  D $
 +
of the complex  $  z $-
 +
plane. Let $  l $
 +
be the maximal connected component of the level line
  
<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/a/a011/a011210/a011210110.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Re}  \int\limits _ {z _ {0} } ^ { z }  \sqrt {q (t) } \
 +
d t  = 0 ,
 +
$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a011210111.png" />.
+
emanating from a turning point  $  z _ {0} $
 +
and containing no other turning points; then  $  l $
 +
is called a Stokes line. If  $  q = - z $(
 +
that is, (4) is the Airy equation), then the Stokes lines are the rays  $  ( - \infty , 0 ) $
 +
and  $  ( 0 , e ^ {\pm i \pi / 3 } ) $.
 +
Analogously, if  $  z _ {0} $
 +
is a simple turning point of (4), then there are three Stokes lines  $  l _ {1} , l _ {2} $
 +
and  $  l _ {3} $
 +
emanating from it and the angle between adjacent lines at  $  z _ {0} $
 +
is equal to  $  2 \pi / 3 $.  
 +
Let  $  S _ {j} $
 +
be a neighbourhood of  $  z _ {0} $
 +
from which a neighbourhood of the Stokes line  $  l _ {j} $,
 +
$  j = 1 , 2 , 3 $,
 +
has been removed. For a suitable numbering of the  $  S _ {j} $,
 +
equation (4) has three solutions  $  \widetilde{w}  _ {j} (z) $,
 +
$  j = 1 , 2 , 3 $,
 +
such that, as  $  \lambda \rightarrow + \infty $,
 +
 
 +
$$
 +
\widetilde{w}  _ {j} (z)  \sim 
 +
\frac{1}{\sqrt {\xi (z) } }
 +
v ( - \lambda
 +
^ {2/3} \omega  ^ {j} \xi (z) ) ,\ \
 +
\omega  =  e ^ {2 \pi i / 3 } ,
 +
$$
 +
 
 +
for  $  z \in S _ {j} $.
  
 
The Airy functions also occur in the study of asymptotic solutions of ordinary differential equations and systems of higher order near simple turning points.
 
The Airy functions also occur in the study of asymptotic solutions of ordinary differential equations and systems of higher order near simple turning points.
Line 118: Line 353:
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.B. Airy,  ''Trans. Cambridge Philos. Soc.'' , '''6'''  (1838)  pp. 379–402</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.G. Stokes,  ''Trans. Cambridge Philos. Soc.'' , '''10'''  (1857)  pp. 105–128</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.A. Fok,  "Tables of the Airy functions" , Moscow  (1946)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Segun,  M. Abramowitz,  "Handbook of mathematical functions" , ''Appl. Math. Ser.'' , '''55''' , Nat. Bur. Standards  (1970)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.M. Babich,  V.S. Buldyrev,  "Asymptotic methods in the diffraction of short waves" , Moscow  (1972)  (In Russian)  (Translation forthcoming: Springer)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M.V. Fedoryuk,  "The saddle-point method" , Moscow  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E.M. Lifshits,  "The classical theory of fields" , Addison-Wesley  (1951)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  V.P. Maslov,  M.V. Fedoryuk,  "Quasi-classical approximation for the equations of quantum mechanics" , Reidel  (1981)  (Translated from Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A.A. Dorodnitsyn,  "Asymptotic laws of distribution of the characteristic values for certain types of second-order differential equations"  ''Uspekhi Mat. Nauk'' , '''6''' :  7  (1952)  pp. 3–96  (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  W. Wasov,  "Asymptotic expansions for ordinary differential equations" , Interscience  (1965)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  M.V. Fedoryuk,  "Asymptotic methods for linear ordinary differential equations" , Moscow  (1983)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.B. Airy,  ''Trans. Cambridge Philos. Soc.'' , '''6'''  (1838)  pp. 379–402</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.G. Stokes,  ''Trans. Cambridge Philos. Soc.'' , '''10'''  (1857)  pp. 105–128</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.A. Fok,  "Tables of the Airy functions" , Moscow  (1946)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Segun,  M. Abramowitz,  "Handbook of mathematical functions" , ''Appl. Math. Ser.'' , '''55''' , Nat. Bur. Standards  (1970)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.M. Babich,  V.S. Buldyrev,  "Asymptotic methods in the diffraction of short waves" , Moscow  (1972)  (In Russian)  (Translation forthcoming: Springer)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M.V. Fedoryuk,  "The saddle-point method" , Moscow  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E.M. Lifshits,  "The classical theory of fields" , Addison-Wesley  (1951)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  V.P. Maslov,  M.V. Fedoryuk,  "Quasi-classical approximation for the equations of quantum mechanics" , Reidel  (1981)  (Translated from Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A.A. Dorodnitsyn,  "Asymptotic laws of distribution of the characteristic values for certain types of second-order differential equations"  ''Uspekhi Mat. Nauk'' , '''6''' :  7  (1952)  pp. 3–96  (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  W. Wasov,  "Asymptotic expansions for ordinary differential equations" , Interscience  (1965)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  M.V. Fedoryuk,  "Asymptotic methods for linear ordinary differential equations" , Moscow  (1983)  (In Russian)</TD></TR></table>
  
 +
====Comments====
 +
The Airy function can be expressed in terms of modified [[Bessel functions|Bessel functions]] of the third kind:
  
 +
$$
 +
\mathop{\rm Ai} (x)  = \
  
====Comments====
+
\frac{1}{\pi \sqrt 3 }
The Airy function can be expressed in terms of modified [[Bessel functions|Bessel functions]] of the third kind:
+
 
 +
\sqrt x K _ {1/3} \left (
  
<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/a/a011/a011210/a011210112.png" /></td> </tr></table>
+
\frac{2}{3}
 +
x  ^ {2/3}
 +
\right ) .
 +
$$
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a011210113.png" /> satisfies the differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011210/a011210114.png" />, cf. [[#References|[a2]]].
+
The function $  \mathop{\rm Ai} (z) $
 +
satisfies the differential equation $  w  ^ {\prime\prime} (z) = z w (z) $,  
 +
cf. [[#References|[a2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.W.J. Olver,  "Asymptotics and special functions" , Acad. Press  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.N. Lebedev,  "Special functions and their applications" , Dover, reprint  (1972)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.W.J. Olver,  "Asymptotics and special functions" , Acad. Press  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.N. Lebedev,  "Special functions and their applications" , Dover, reprint  (1972)  (Translated from Russian)</TD></TR></table>

Latest revision as of 16:09, 1 April 2020


Particular solutions of the Airy equation.

The first Airy function (or simply the Airy function) is defined by

$$ \mathop{\rm Ai} (x) = \frac{1} \pi \int\limits _ { 0 } ^ \infty \cos \left ( \frac{t ^ {3} }{3} + x t \right ) \ d t . $$

For complex values of $ z $

$$ \mathop{\rm Ai} (z) = \frac{1}{2 \pi } \int\limits _ \gamma \mathop{\rm exp} \left ( zt - \frac{t ^ {3} }{3} \right ) \ d t , $$

where $ \gamma = ( \infty e ^ {- 2 \pi i / 3 } , 0 ] \cup [ 0 , + \infty ) $ is a contour in the complex $ t $- plane. The second Airy function is defined by

$$ \mathop{\rm Bi} (z) = i \omega ^ {2} \mathop{\rm Ai} ( \omega ^ {2} z ) - i \omega \mathop{\rm Ai} ( \omega z ) ,\ \omega = e ^ {2 \pi i / 3 } . $$

The functions $ \mathop{\rm Ai} (x) $ and $ \mathop{\rm Bi} (x) $ are real for real $ x $.

A second collection of Airy functions was introduced by V.A. Fock [V.A. Fok]:

$$ v (z) = \frac{\sqrt \pi }{2} \mathop{\rm Ai} (z) , $$

$$ w _ {1} (z) = 2 e ^ {i \pi / 6 } v ( \omega z ) , $$

$$ w _ {2} (z) = 2 e ^ {- i \pi / 6 } v ( \omega ^ {-1} z ) ; $$

in this case $ v (z) $ is called the Airy–Fok function (Airy–Fock function). The following identities hold:

$$ \tag{1 } v (z) = \frac{w _ {1} (z) - w _ {2} (z) }{2i} ,\ \ \overline{ {w _ {1} (z) }}\; = w _ {2} ( \overline{z}\; ) . $$

Any two of $ v (z) , w _ {1} (z) $ and $ w _ {2} (z) $ are linearly independent.

The most important Airy function is $ v (z) $( or $ \mathop{\rm Ai} (z) $). Its asymptotic behaviour on the real axis is given by

$$ v (x) = \frac{1}{2} \frac{x ^ {-1/4} }{\sqrt \pi } \ \mathop{\rm exp} \left ( - \frac{2}{3} x ^ {3/2} \right ) [ 1 + O ( x ^ {-3/2} ) ] , \ x \rightarrow + \infty , $$

$$ v (x) = \frac{| x | ^ {-1/4} }{\sqrt \pi } \left [ \sin \left ( \frac{2}{3} | x | ^ {3/2} + \frac \pi {4} \right ) + O ( | x | ^ {-3/2} ) \right ] , $$

$$ \ x \rightarrow - \infty , $$

so $ v (x) $ decreases rapidly for $ x > 0 , x \gg 1 $ and oscillates strongly for $ x < 0 , | x | \gg 1 $. The functions $ w _ {1} (x) $ and $ w _ {2} (x) $ increase exponentially as $ x \rightarrow + \infty $. For complex $ z $ the Airy functions have the following asymptotic expansions as $ | z | \rightarrow \infty $:

$$ \tag{2 } v (z) \sim \frac{1}{2 \sqrt \pi } z ^ {-1/4} \mathop{\rm exp} \left ( - \frac{2}{3} z ^ {3/2} \right ) \times $$

$$ \times \sum _ { n=0 } ^ \infty (-1) ^ {n} a _ {n} z ^ {-3n/2} \ \textrm{ for } | \mathop{\rm arg} z | \leq \pi - \epsilon , $$

$$ w _ {1} (z) \sim \frac{1}{\sqrt \pi } z ^ {-1/4} \mathop{\rm exp} \left ( \frac{2}{3} z ^ {3/2} \right ) \times $$

$$ \times \sum _ { n=0 } ^ \infty a _ {n} z ^ {-3n/2} \ \textrm{ for } \ | \mathop{\rm arg} z - \frac \pi {3} | \leq \pi - \epsilon , $$

where

$$ a _ {n} = \ \frac{\Gamma \left ( 3 n + \frac{1}{2} \right ) 9 ^ {-n} }{( 2 n ) ! } . $$

The asymptotic expansion of $ w _ {2} (z) $ is of the form (2), but it is valid in the sector

$$ \left | \mathop{\rm arg} \left ( z + \frac \pi {3} \right ) \ \right | \leq \pi - \epsilon . $$

Here $ \epsilon \in ( 0 , \pi ) $ is arbitrary, the branches of $ \sqrt z $ and $ z ^ {1/4} $ are positive on the semi-axis $ ( 0 , \infty ) $, and the asymptotic expansions are uniform with respect to $ \mathop{\rm arg} z $ and can be differentiated term by term any number of times. In the remaining sector $ | \mathop{\rm arg} -z | < \epsilon $ the asymptotic expansion of $ v (z) $ is expressed in terms of those of $ w _ {1} (z) $ and $ w _ {2} (z) $ by means of (1); hence, the asymptotic expansion of $ v (z) $ has a different form in different sectors of the complex $ z $- plane. This fact was first established by G.G. Stokes [2] and is called the Stokes phenomenon.

The Airy functions occur in the study of integrals of rapidly-oscillating functions, of the form

$$ I ( \lambda , \alpha ) = \int\limits _ { a } ^ { b } \textrm{ e } ^ {i \lambda S ( x , \alpha ) } f ( x , \alpha ) d x , $$

for $ \lambda > 0 , \lambda \rightarrow \infty $. Here $ f $ and $ S $ are smooth functions, $ S $ is real and $ \alpha $ is a real parameter. If for small values of $ \alpha \geq 0 $ the phase $ S $ has two close non-degenerate stationary points $ x _ {1} ( \alpha ) $ and $ x _ {2} ( \alpha ) $ that coincide for $ \alpha = 0 $, for example, if

$$ S ( x , \alpha ) = \alpha x - x ^ {3} + O ( x ^ {4} ) \ \textrm{ as } x \rightarrow 0 , $$

then for small values of $ \alpha \geq 0 $, as $ \lambda \rightarrow + \infty $, the contribution to the asymptotics of the integral coming from a neighbourhood of the point $ x = 0 $ can be expressed in terms of the Airy function $ v $ and its derivative (see [6]). Integrals of this kind occur in the study of short-wave fields near a simple focus (see [7] and [8]); the Airy functions arose in connection with the study of this problem [1].

Consider the second-order differential equation

$$ \tag{3 } y ^ {\prime\prime} + \lambda ^ {2} q (x) y = 0 , $$

where $ q (x) $ is a smooth real-valued function on the interval $ I = [ a , b ] $ and $ \lambda > 0 $ is a large parameter. The zeros of $ q (x) $ are called turning points (or transfer points) of the equation (3). Let

$$ a < x _ {0} < b ,\ \ q ( x _ {0} ) = 0 ,\ \ q ^ \prime ( x _ {0} ) \neq 0 $$

(such a point is called simple),

$$ q (x) \neq 0 \ \textrm{ for } \ x \in I , x \neq x _ {0} ,\ \ q ^ \prime ( x _ {0} ) > 0 . $$

Set

$$ \xi (x) = \left ( \frac{2}{3} \int\limits _ {x _ {0} } ^ { x } \sqrt {q (t) } d t \right ) ^ {2/3} ,\ \ \mathop{\rm sign} \xi (x) = \mathop{\rm sign} ( x - x _ {0} ) , $$

$$ Y _ {0} (x) = ( \xi ^ \prime (x) ) ^ {-1/2} \mathop{\rm Ai} ( - \lambda ^ {2/3} \xi (x) ) , $$

$$ Y _ {1} (x) = ( \xi ^ \prime (x) ) ^ {-1/2} \mathop{\rm Bi} ( - \lambda ^ {2/3} \xi (x) ) . $$

Equation (3) has linearly independent solutions $ y _ {0} (x) $ and $ y _ {1} (x) $ such that, as $ \lambda \rightarrow + \infty $,

$$ y _ {j} (x) = Y _ {j} (x) \left [ 1 + O \left ( \frac{1} \lambda \right ) \right ] ,\ \ a \leq x \leq x _ {0} ,\ j = 0 , 1 , $$

$$ y _ {0} (x) = Y _ {0} (x) \left [ 1 + O \left ( \frac{1} \lambda \right ) \right ] + Y _ {1} (x) O \left ( \frac{1} \lambda \right ) , $$

$$ y _ {1} (x) = Y _ {1} (x) \left [ 1 + O \left ( \frac{1} \lambda \right ) \right ] + Y _ {0} (x) O \left ( \frac{1} \lambda \right ) , $$

$$ x _ {0} \leq x \leq b , $$

uniformly with respect to $ x $.

This result has been generalized in various directions: asymptotic series have been obtained for the solutions, the case $ q = q ( x , \lambda ) $ has been studied (for example, if $ q ( x , \lambda ) $ can be expanded in an asymptotic series $ q \sim \sum _ {n=0} ^ \infty \lambda ^ {-n} q _ {n} (x) $ as $ \lambda \rightarrow + \infty $), and the asymptotic behaviour of the solutions near multiple turning points has been investigated. Other generalizations concern the equation

$$ \tag{4 } w ^ {\prime\prime} + \lambda ^ {2} q (x) w = 0 , $$

where the function $ q (z) $ is analytic in a domain $ D $ of the complex $ z $- plane. Let $ l $ be the maximal connected component of the level line

$$ \mathop{\rm Re} \int\limits _ {z _ {0} } ^ { z } \sqrt {q (t) } \ d t = 0 , $$

emanating from a turning point $ z _ {0} $ and containing no other turning points; then $ l $ is called a Stokes line. If $ q = - z $( that is, (4) is the Airy equation), then the Stokes lines are the rays $ ( - \infty , 0 ) $ and $ ( 0 , e ^ {\pm i \pi / 3 } ) $. Analogously, if $ z _ {0} $ is a simple turning point of (4), then there are three Stokes lines $ l _ {1} , l _ {2} $ and $ l _ {3} $ emanating from it and the angle between adjacent lines at $ z _ {0} $ is equal to $ 2 \pi / 3 $. Let $ S _ {j} $ be a neighbourhood of $ z _ {0} $ from which a neighbourhood of the Stokes line $ l _ {j} $, $ j = 1 , 2 , 3 $, has been removed. For a suitable numbering of the $ S _ {j} $, equation (4) has three solutions $ \widetilde{w} _ {j} (z) $, $ j = 1 , 2 , 3 $, such that, as $ \lambda \rightarrow + \infty $,

$$ \widetilde{w} _ {j} (z) \sim \frac{1}{\sqrt {\xi (z) } } v ( - \lambda ^ {2/3} \omega ^ {j} \xi (z) ) ,\ \ \omega = e ^ {2 \pi i / 3 } , $$

for $ z \in S _ {j} $.

The Airy functions also occur in the study of asymptotic solutions of ordinary differential equations and systems of higher order near simple turning points.

References

[1] G.B. Airy, Trans. Cambridge Philos. Soc. , 6 (1838) pp. 379–402
[2] G.G. Stokes, Trans. Cambridge Philos. Soc. , 10 (1857) pp. 105–128
[3] V.A. Fok, "Tables of the Airy functions" , Moscow (1946) (In Russian)
[4] A. Segun, M. Abramowitz, "Handbook of mathematical functions" , Appl. Math. Ser. , 55 , Nat. Bur. Standards (1970)
[5] V.M. Babich, V.S. Buldyrev, "Asymptotic methods in the diffraction of short waves" , Moscow (1972) (In Russian) (Translation forthcoming: Springer)
[6] M.V. Fedoryuk, "The saddle-point method" , Moscow (1977) (In Russian)
[7] E.M. Lifshits, "The classical theory of fields" , Addison-Wesley (1951) (Translated from Russian)
[8] V.P. Maslov, M.V. Fedoryuk, "Quasi-classical approximation for the equations of quantum mechanics" , Reidel (1981) (Translated from Russian)
[9] A.A. Dorodnitsyn, "Asymptotic laws of distribution of the characteristic values for certain types of second-order differential equations" Uspekhi Mat. Nauk , 6 : 7 (1952) pp. 3–96 (In Russian)
[10] W. Wasov, "Asymptotic expansions for ordinary differential equations" , Interscience (1965)
[11] M.V. Fedoryuk, "Asymptotic methods for linear ordinary differential equations" , Moscow (1983) (In Russian)

Comments

The Airy function can be expressed in terms of modified Bessel functions of the third kind:

$$ \mathop{\rm Ai} (x) = \ \frac{1}{\pi \sqrt 3 } \sqrt x K _ {1/3} \left ( \frac{2}{3} x ^ {2/3} \right ) . $$

The function $ \mathop{\rm Ai} (z) $ satisfies the differential equation $ w ^ {\prime\prime} (z) = z w (z) $, cf. [a2].

References

[a1] F.W.J. Olver, "Asymptotics and special functions" , Acad. Press (1974)
[a2] N.N. Lebedev, "Special functions and their applications" , Dover, reprint (1972) (Translated from Russian)
How to Cite This Entry:
Airy functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Airy_functions&oldid=17747
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article