# Stationary phase, method of the

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 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 ).

If $x ^ {0} \in \Omega$ is a stationary point of finite multiplicity, then (see )

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

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

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 (see , ). 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 $B$. If the functions are analytic, then the method of the stationary phase is a particular case of the saddle point method.

How to Cite This Entry:
Stationary phase, method of the. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stationary_phase,_method_of_the&oldid=52173
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article