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A method for calculating the asymptotics of integrals of rapidly-oscillating functions:

$$ \tag{* }
F( \lambda )  =  \int\limits _  \Omega  f( x) e ^ {i \lambda S( x) }  dx,
$$

where  $  x \in \mathbf R  ^ {n} $, 
$  \lambda > 0 $, 
$  \lambda \rightarrow + \infty $, 
is a large parameter,  $  \Omega $
is a bounded domain, the function  $  S( x) $(
the phase) is real, the function  $  f( x) $
is complex, and  $  f, S \in C  ^  \infty  ( \mathbf R  ^ {n} ) $. 
If  $  f \in C _ {0}  ^  \infty  ( \mathbf R  ^ {n} ) $, 
i.e.  $  f $
has compact support, and the phase  $  S( x) $
does not have stationary points (i.e. points at which  $  S  ^  \prime  ( x) = 0 $) 
on  $  \supp  f $, 
$  \Omega = \mathbf R  ^ {n} $, 
then  $  F( \lambda ) = O( \lambda ^ {- n } ) $, 
for all  $  n $
as  $  \lambda \rightarrow + \infty $. 
Therefore, when  $  \lambda \rightarrow + \infty $, 
the points of stationary phase and the boundary  $  \partial  \Omega $
give the essential contribution to the asymptotics of the integral (*). The integrals

$$ 
V _ {x  ^ {0}  } ( \lambda )  = \ 
\int\limits _  \Omega  f( x) \phi _ {0} ( x) e ^ {i \lambda S( x) }  dx ,
$$

$$ 
V _ {\partial  \Omega }  ( \lambda )  =  \int\limits _  \Omega 
f( x) \phi _ {\partial  \Omega }  ( x) e ^ {i \lambda S( x) }  dx
$$

are called the contributions from the isolated stationary point  $  x  ^ {0} $
and the boundary, respectively, where  $  \phi _ {0} \in C _ {0}  ^  \infty  ( \Omega ) $, 
$  \phi _ {0} \equiv 1 $
near the point  $  x  ^ {0} $
and  $  \supp  \phi _ {0} $
does not contain any other stationary points,  $  \phi _ {\partial  \Omega }  \in C _ {0}  ^  \infty  ( \mathbf R  ^ {n} ) $
and  $  \phi _ {\partial  \Omega }  \equiv 1 $
in a certain neighbourhood of the boundary. For  $  n= 1 $, 
$  \Omega = ( a, b) $:

1)  $  V _ {a} ( \lambda ) = 
\frac{i}{\lambda S  ^  \prime  ( a) }
 e ^ {i \lambda S( a) } [ f( a) + O( \lambda  ^ {-} 1 )] $, 
if  $  S  ^  \prime  ( a) \neq 0 $;

2)

$$ 
V _ {x  ^ {0}  } ( \lambda )  =  \sqrt {
\frac{2 \pi }{\lambda | S  ^ {\prime\prime} ( x  ^ {0} ) | }
 } e ^ {i ( \lambda S ( x  ^ {0} ) + \pi \delta _ {0} / 4 ) } \times
$$

$$ 
\times 
[ f( x  ^ {0} ) + O( \lambda  ^ {-} 1 )],\ \ 
\delta _ {0} =  \mathop{\rm sgn}  S  ^ {\prime\prime} ( x  ^ {0} ),
$$

if  $  x  ^ {0} $
is an interior point of  $  \Omega $
and  $  S  ^  \prime  ( x  ^ {0} ) = 0 $, 
$  S  ^ {\prime\prime} ( x  ^ {0} ) \neq 0 $.

Detailed research has been carried out in the case where  $  n= 1 $, 
the phase  $  S $
has a finite number of stationary points, all of finite multiplicity, and the function  $  f $
has zeros of finite multiplicity at these points and at the end-points of an interval  $  \Omega $. 
Asymptotic expansions have been obtained. The case where the functions  $  f $
and  $  S $
have power singularities has also been studied: for example,  $  f = x  ^  \alpha  f _ {1} ( x) $, 
$  S = x  ^  \beta  S _ {1} ( x) $, 
where  $  f _ {1} $, 
$  S _ {1} $
are smooth functions when  $  x = 0 $, 
$  \alpha > - 1 $, 
$  \beta > 0 $.

Let  $  n \geq  2 $, 
and let  $  x  ^ {0} \in \Omega $
be a non-degenerate stationary point (i.e.  $  \Delta _ {S} ( x  ^ {0} ) =  \mathop{\rm det}  S  ^ {\prime\prime} ( x  ^ {0} ) \neq 0 $). 
The contribution from the point  $  x  ^ {0} $
is then equal to

$$ 
V _ {x  ^ {0}  } ( \lambda )  = \ 
\left ( 
\frac{2 \pi } \lambda 
 \right )  ^ {n/2} | \Delta _ {S} ( x  ^ {0} ) |  ^ {-} 1/2 \times
$$

$$ 
\times 
 \mathop{\rm exp} \left [ i \left ( \lambda S( x  ^ {0} ) + + 
\frac \pi {4}

\delta _ {S} ( x  ^ {0} ) \right ) \right ] [ f( x  ^ {0} ) + O( \lambda  ^ {-} 1 )],
$$

where  $  \delta _ {S} ( x  ^ {0} ) $
is the [[Signature|signature]] of the matrix  $  S  ^ {\prime\prime} ( x  ^ {0} ) $. 
There is also an asymptotic series for  $  V _ {x  ^ {0}  } ( \lambda ) $(
for the formulas of the contribution  $  V _ {\partial  \Omega }  ( \lambda ) $
in the case of a smooth boundary, see [[#References|[5]]]).

If  $  x  ^ {0} \in \Omega $
is a stationary point of finite multiplicity, then (see [[#References|[6]]])

$$ 
V _ {x  ^ {0}  } ( \lambda )  \sim   \mathop{\rm exp} [ i \lambda S( x  ^ {0} )] \sum _ { k= } 0 ^  \infty   \left ( \sum _ { l= } 0 ^ { N }  a _ {kl} \lambda ^ {- r _ {k} } (  \mathop{\rm ln}  \lambda )  ^ {l} \right ) ,
$$

where  $  r _ {k} $
are rational numbers,  $  n/2 \leq  r _ {0} < \dots < r _ {k} \rightarrow + \infty $. 
Degenerate stationary points have been studied, cf. [[#References|[3]]], [[#References|[4]]].

Studies have been made on the case where the phase  $  S = S( x, \alpha ) $
depends on a real parameter  $  \alpha $, 
and for small  $  | \alpha | $
has two close non-degenerate stationary points. In this case, the asymptotics of the integral  $  F( \lambda , \alpha ) $
can be expressed in terms of [[Airy functions|Airy functions]] (see [[#References|[5]]], [[#References|[10]]]). The method of the stationary phase has an operator variant:  $  \lambda = A $, 
where  $  A $
is the infinitesimal operator of the strongly-continuous group  $  \{ e  ^ {itA} \} $
of operators bounded on the axis  $  - \infty < t < \infty $, 
acting on a Banach space  $  B $, 
and  $  f( x) $, 
$  S( x) $
are smooth functions with values in  $  B $[[#References|[9]]]. If the functions are analytic, then the method of the stationary phase is a particular case of the [[Saddle point method|saddle point method]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> W. Thomson, ''Philos. Mag.'' , '''23''' (1887) pp. 252–255</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Erdélyi, "Asymptotic expansions" , Dover, reprint (1956) {{MR|0081379}} {{MR|0078494}} {{ZBL|0072.11703}} {{ZBL|0070.29002}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.Ya. Rieksteyn'sh, "Asymptotic expansions of integrals" , '''1–2''' , Riga (1974–1977) (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> F.W.J. Olver, "Asymptotics and special functions" , Acad. Press (1974) {{MR|0435697}} {{ZBL|0308.41023}} {{ZBL|0303.41035}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M.V. Fedoryuk, "The method of steepest descent" , Moscow (1977) (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> M.F. Atiyah, "Resolution of singularities and division of distributions" ''Comm. Pure Appl. Math.'' , '''23''' : 2 (1970) pp. 145–150 {{MR|0256156}} {{ZBL|0188.19405}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> V.I. Arnol'd, "Remarks on the stationary phase method and Coxeter numbers" ''Russian Math. Surveys'' , '''28''' : 5 (1973) pp. 19–48 ''Uspekhi Mat. Nauk'' , '''28''' : 5 (1973) pp. 17–44 {{MR|}} {{ZBL|0291.40005}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.N. Varchenko, "Newton polyhedra and estimation of oscillating integrals" ''Funct. Anal. Appl'' , '''10''' : 3 (1976) pp. 175–196 ''Funktsional. Anal. i Prilozhen.'' , '''10''' : 3 (1976) pp. 13–38 {{MR|}} {{ZBL|0351.32011}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> V.P. Maslov, M.V. Fedoryuk, "Semi-classical approximation in quantum mechanics" , Reidel (1981) (Translated from Russian) {{MR|}} {{ZBL|0458.58001}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> M.V. Fedoryuk, "Asymptotics. Integrals and series" , Moscow (1987) (In Russian) {{MR|0950167}} {{ZBL|0641.41001}} </TD></TR></table>

====Comments====
An integral of the form (*) is a special case of a so-called oscillatory integral, or [[Fourier integral operator|Fourier integral operator]], cf. also [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Wong, "Asymptotic approximations of integrals" , Acad. Press (1989) {{MR|1016818}} {{ZBL|0679.41001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> L.V. Hörmander, "The analysis of linear partial differential operators" , '''1''' , Springer (1983) pp. §7.7 {{MR|0717035}} {{MR|0705278}} {{ZBL|0521.35002}} {{ZBL|0521.35001}} </TD></TR></table>