In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the stationary phase approximation is a basic principle of
asymptotic analysis
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.
As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as beco ...
, applying to the limit as
.
This method originates from the 19th century, and is due to
George Gabriel Stokes
Sir George Gabriel Stokes, 1st Baronet, (; 13 August 1819 – 1 February 1903) was an Irish migration to Great Britain, Irish English physicist and mathematician. Born in County Sligo, Ireland, Stokes spent all of his career at the University ...
and
Lord Kelvin
William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, Mathematical physics, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy (Glasgow), Professor of Natural Philoso ...
.
It is closely related to
Laplace's method
In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form
:\int_a^b e^ \, dx,
where f(x) is a twice-differentiable function, ''M'' is a large number, and the endpoints ''a'' an ...
and the
method of steepest descent
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in r ...
, but Laplace's contribution precedes the others.
Basics
The main idea of stationary phase methods relies on the cancellation of sinusoids with rapidly varying phase. If many sinusoids have the same phase and they are added together, they will add constructively. If, however, these same sinusoids have phases which change rapidly as the frequency changes, they will add incoherently, varying between constructive and destructive addition at different times.
Formula
Letting
denote the set of
critical points of the function
(i.e. points where
), under the assumption that
is either compactly supported or has exponential decay, and that all critical points are nondegenerate (i.e.
for
) we have the following asymptotic formula, as
:
:
Here
denotes the
Hessian
A Hessian is an inhabitant of the German state of Hesse.
Hessian may also refer to:
Named from the toponym
*Hessian (soldier), eighteenth-century German regiments in service with the British Empire
**Hessian (boot), a style of boot
**Hessian f ...
of
, and
denotes the
signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
of the Hessian, i.e. the number of positive eigenvalues minus the number of negative eigenvalues.
For
, this reduces to:
:
In this case the assumptions on
reduce to all the critical points being non-degenerate.
This is just the
Wick-rotated version of the formula for the
method of steepest descent
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in r ...
.
An example
Consider a function
:
.
The phase term in this function,
, is stationary when
:
or equivalently,
:
.
Solutions to this equation yield dominant frequencies
for some
and
. If we expand
as a
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
about
and neglect terms of order higher than
, we have
:
where
denotes the second derivative of
. When
is relatively large, even a small difference
will generate rapid oscillations within the integral, leading to cancellation. Therefore we can extend the limits of integration beyond the limit for a Taylor expansion. If we use the formula,
:
.
:
.
This integrates to
: