Method of Stationary Phase

where

(910) |

and

(911) |

Now, is a relatively slowly varying function of (except in the immediate vicinity of the singular points, ), whereas the phase is generally large and rapidly varying. The rapid oscillations of over most of the range of integration means that the integrand averages to almost zero. Exceptions to this cancellation rule occur only at points where is stationary: that is, where has an extremum. The integral can therefore be estimated by finding all the points in the -plane where has a vanishing derivative, evaluating (approximately) the integral in the neighborhood of each of these points, and summing the contributions. This procedure is known as the

Suppose that has a vanishing first derivative at . In the neighborhood of this point, can be expanded as a Taylor series,

(912) |

Here, the subscript is used to indicate , or its second derivative, evaluated at , whereas denotes a derivative with respect to . Because is slowly varying, the contribution to the integral from this stationary phase point is approximately

(913) |

It is tacitly assumed that the stationary point lies on the real axis in -space, so that locally the integral along the contour is an integral along the real axis in the direction of decreasing . The previous expression can be written in the form

(914) |

where

(915) |

The integrals in the previous expression are known as

(916) |

It follows that

It is easily demonstrated that the arc-length (in the -plane) of the section of the integration contour that makes a significant contribution to is of order . Thus, the arc-length is relatively short, provided that the wavelength of the signal is much less than the distance it has propagated into the dispersive medium. If there is more than one point of stationary phase in the range of integration then the integral is approximated as a sum of terms having the same form as the previous one.

Integrals of the form (910) can be calculated exactly using the
*method of steepest decent*.^{} The stationary
phase approximation (918) agrees with the leading term of the
method of steepest decent (which is far more difficult to implement
than the method of stationary phase) provided that
is
real (i.e., provided that
the stationary point lies on the real axis). If
is complex, however, then the stationary phase
method can yield erroneous results. This suggests that the stationary
phase method is likely to break down when the extremum point
approaches any poles or branch cuts in the
-plane.