# Difference between revisions of "Method of stationary phase"

Many useful results in mathematical physics are the result of asymptotic approximations. That is, the are formulae and results that are obtained by deriving an asymptotic series and keeping the leading order term as the approximation, and using the next order term as an error estimate.[1] [2][3]

On such result is called The method of stationary phase, which applies to integrals that resemble the Fourier transform, but have a more general phase function. The method of stationary phase applies to one dimensional Fourier-like integrals as well as to multi-dimensional Fourier integrals.

Many wave phenomena are the result of the preferential constructive interference of the wavefield. These phenomena are examples of result that would be described by extrema of the phase function of a Fourier representation.

## The method of stationary phase in 1-dimension

We consider integrals of the form

${\displaystyle I(\lambda )=\int _{a}^{b}f(t)e^{i\lambda \phi (t)}\;dt}$.

Here, ${\displaystyle \lambda >0}$ is a large parameter, which may be frequency or wave number in problems of wave propagation. The function ${\displaystyle f(x)}$ is called the amplitude and the real-valued function ${\displaystyle \phi (x)}$ is called the phase. As the name of the technique implies our points of interest are places where the phase function is slowly varying.

### Simple critical point

The endpoints of integration, places where the derivatives of ${\displaystyle f(t)}$ fail to be continuous, and places where the derivatives of ${\displaystyle \phi (t)}$ vanish are called critical points.

A simple critical point is a point ${\displaystyle t=a}$, where ${\displaystyle \phi ^{\prime }(a)=0}$ but ${\displaystyle \phi ^{\prime \prime }(a)\neq 0}$. Such a simple critical point is also called a stationary point because this is the place where the phase function has a minimum or a maximum and is thus, stationary.

Through a number of coordinate transformations, the Fourier-like integral can be repetitively integrated by parts to yield an asymptotic series with large parameter ${\displaystyle \lambda }$. The leading order term of that series is called the stationary phase formula and the second term or third term is the estimate of the asymptotic error of the approximation.

We consider three possibilities, that the stationary point is on the lower endpoint of integration ${\displaystyle t=a}$, the upper endpoint of integration ${\displaystyle t=b}$, or an interior stationary point ${\displaystyle t=c}$, where ${\displaystyle a.

#### Stationary phase formula -- stationary point at the lower endpoint of integration ${\displaystyle t=a}$

${\displaystyle I_{a}(\lambda )\sim {\frac {1}{2}}e^{i\lambda \phi (a)+i\operatorname {sgn} \left(\phi ^{\prime \prime }(a)\right)\pi /4}\left\{f(a){\sqrt {\frac {2\pi }{\lambda |\phi ^{\prime \prime }(a)|}}}+{\frac {2}{\lambda |\phi ^{\prime \prime }(a)|}}\left[f^{\prime }(a)-{\frac {\phi ^{\prime \prime \prime }(a)}{3|\phi ^{\prime \prime }(a)|}}\right]e^{i\lambda \operatorname {sgn}(\phi ^{\prime \prime }(a))\pi /4}\right\}+O(\lambda ^{-3/2})}$

In practice the stationary phase formula for a stationary point at the lower limit of integration is the leading order term in inverse powers of ${\displaystyle \lambda }$

${\displaystyle I_{a}(\lambda )\sim {\frac {1}{2}}e^{i\lambda \phi (a)+i\operatorname {sgn} \left(\phi ^{\prime \prime }(a)\right)\pi /4}f(a){\sqrt {\frac {2\pi }{\lambda |\phi ^{\prime \prime }(a)|}}}+O(\lambda ^{-1})}$,

as ${\displaystyle \lambda \rightarrow \infty }$.

#### Stationary phase formula -- upper endpoint of integration ${\displaystyle t=b}$

${\displaystyle I_{b}(\lambda )\sim {\frac {1}{2}}e^{i\lambda \phi (b)+i\operatorname {sgn} \left(\phi ^{\prime \prime }(b)\right)\pi /4}\left\{f(b){\sqrt {\frac {2\pi }{\lambda |\phi ^{\prime \prime }(b)|}}}-{\frac {2}{\lambda |\phi ^{\prime \prime }(b)|}}\left[f^{\prime }(b)-{\frac {\phi ^{\prime \prime \prime }(b)}{3|\phi ^{\prime \prime }(b)|}}\right]e^{i\lambda \operatorname {sgn}(\phi ^{\prime \prime }(b))\pi /4}\right\}+O(\lambda ^{-3/2})}$

In practice the stationary phase formula for a stationary point at the upper limit of integration is the leading order term in inverse powers of ${\displaystyle \lambda }$

${\displaystyle I_{b}(\lambda )\sim {\frac {1}{2}}e^{i\lambda \phi (b)+i\operatorname {sgn} \left(\phi ^{\prime \prime }(b)\right)\pi /4}f(b){\sqrt {\frac {2\pi }{\lambda |\phi ^{\prime \prime }(b)|}}}+O(\lambda ^{-1})}$,

as ${\displaystyle \lambda \rightarrow \infty }$.

#### Stationary phase formula -- interior stationary point ${\displaystyle t=c}$, where ${\displaystyle a

The stationary phase expansion at an interior stationary point is found by combining the upper and lower limit stationary phase series expansions, which causes the terms of order ${\displaystyle O(\lambda ^{-1})}$ yielding the familiar result

${\displaystyle I_{c}(\lambda )\sim e^{i\lambda \phi (c)+i\operatorname {sgn} \left(\phi ^{\prime \prime }(c)\right)\pi /4}f(a){\sqrt {\frac {2\pi }{\lambda |\phi ^{\prime \prime }(c)|}}}+O(\lambda ^{-3/2})}$,

as ${\displaystyle \lambda \rightarrow \infty }$.

## What about the endpoint contribution?

When we learned to compute definite integrals in undergraduate calculus, we were evaluating the integral at the limits of integration. We call these limits of integration the endpoints. Thus, we are accustomed to evaluating the endpoint contribution. What is asymptotic order of the endpoint contribution compared with the stationarity contribution?

We can answer that question by performing repetitive integration by parts formally to yield the following series representation. Here, we assume that there are no singularities of the amplitude or the phase of our Fourier-like integral

${\displaystyle I(\lambda )=\int _{a}^{b}f(t)e^{i\lambda \phi (t)}\;dt}$

and apply integration by parts repetitively so as to bring down factors of the large parameter ${\displaystyle \lambda }$ in the denominators of the resulting terms. To do this we multiply and divide the integrant by ${\displaystyle i\lambda \phi ^{\prime }(t)}$. The first application of integration by parts (integrating the exponential) yields

${\displaystyle I(\lambda )=\int _{a}^{b}{\frac {f(t)}{(i\lambda \phi ^{\prime }(t))}}(i\lambda \phi ^{\prime }(t))e^{i\lambda \phi (t)}\;dt=\left.{\frac {f(t)}{(i\lambda \phi ^{\prime }(t))}}e^{i\lambda \phi (t)}\right|_{a}^{b}-{\frac {1}{i\lambda }}\int _{a}^{b}{\frac {d}{dt}}\left[{\frac {f(t)}{\phi ^{\prime }(t)}}\right]e^{i\lambda \phi (t)}\;dt}$.

Applying this operation repetitively yields

${\displaystyle I(\lambda )=\left.\sum _{n=0}^{N-1}{\frac {(-1)^{n}e^{i\lambda \phi (t)}}{(i\lambda )^{n+1}}}\left[{\frac {1}{\phi ^{\prime }(t)}}{\frac {d}{dt}}\right]^{n}\left[{\frac {f(t)}{\phi ^{\prime }(t)}}\right]\right|_{a}^{b}+{\frac {(-1)^{N}}{(i\lambda )^{N}}}\int _{a}^{b}e^{i\lambda \phi (t)}{\frac {d}{dt}}\left[{\frac {1}{\phi ^{\prime }(t)}}{\frac {d}{dt}}\right]^{N-1}\left[{\frac {f(t)}{\phi ^{\prime }(t)}}\right]\;dt}$

as ${\displaystyle \lambda \rightarrow \infty }$.

This formal result assumes that all of the parts are sufficiently differentiable, and there are no divisions by zero. The first term of the summation is ${\displaystyle O(\lambda ^{-1})}$ as ${\displaystyle \lambda \rightarrow \infty }$, whereas a stationary point is of an asymptotically more slowly decaying contribution of ${\displaystyle O(\lambda ^{-1/2})}$.

Endpoint contributions occur wherever there are discontinuities in the data. These constitute such phenomena as ringing, or diffraction smiles seen in processed seismic data.

## Higher order stationary points in 1D

Similar formulas may be derived for higher order stationary points. In general, such a stationary point would be represented as critical points in the amplitude ${\displaystyle f(t)}$ and in the phase ${\displaystyle \phi (t)}$.

### Higher order stationary point at the lower endpoint of integration at ${\displaystyle t=a}$.

In general, we may consider the amplitude factor ${\displaystyle f(t)}$ to represented by the more general power series representations

${\displaystyle f(t)=f_{a}(t-a)^{(\gamma -1)}+o((t-a)^{(\gamma -1).}\qquad }$ for ${\displaystyle \qquad \gamma >0}$

and the phase is represented as

${\displaystyle \phi (t)-\phi (a)=\phi _{a}(t-a)^{\alpha }+o((t-a)^{\alpha })\qquad }$ for ${\displaystyle \qquad \alpha >0}$.

Here ${\displaystyle f_{a}}$ and ${\displaystyle \phi _{a}}$ are the first non vanishing coefficients of the power series representations of the functions.

The resulting stationary phase formula is

${\displaystyle I_{a}(\lambda )\sim {\frac {f_{a}\Gamma (\gamma /\alpha )}{\alpha (\lambda |\phi _{a}|)^{\gamma /\alpha }}}e^{i\lambda \phi _{a}+i\pi \operatorname {sgn}(\phi _{a})\gamma /2\alpha }+o(\lambda ^{\gamma /\alpha })}$

as ${\displaystyle \lambda \rightarrow \infty }$.

### Higher order stationary point at the upper endpoint of integration at ${\displaystyle t=b}$.

In general, we may consider the amplitude factor ${\displaystyle f(t)}$ to represented by the more general power series representations

${\displaystyle f(t)=f_{b}(b-t)^{\beta }+o((b-t)^{\beta })\qquad }$ for ${\displaystyle \qquad \beta >0}$

and the phase is represented as

${\displaystyle \phi (t)-\phi (b)=\phi _{b}(b-t)^{(\delta -1)}+o((b-t)^{(\delta -1)})\qquad }$ for ${\displaystyle \qquad \delta >0}$.

The resulting stationary phase formula is

${\displaystyle I_{b}(\lambda )\sim {\frac {f_{b}\Gamma (\delta /\beta )}{\beta (\lambda |\phi _{b}|)^{\delta /\beta }}}e^{i\lambda \phi _{b}+i\pi \operatorname {sgn}(\phi _{b})\delta /2\beta }+o(\lambda ^{\delta /\beta })}$

as ${\displaystyle \lambda \rightarrow \infty }$.

Here, the Gamma function is

${\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\;dt.}$

## Stationary phase analysis

A common misunderstanding of the method of stationary phase is that this merely a formula lookup. A better approach is to consider that applying the method of stationary phase is a method of analysis. To perform this analysis, given a Fourier-like integral, the following steps must be applied

• identify the large parameter
• identify the phase function, it's derivative, and next highest order non-vanishing derivative at the zeros of the first derivative of the phase. If this is the second derivative of the phase, then this is a simple critical point, also known as a stationary point.
• find the stationary point(s)
• apply the appropriate stationary phase formula.

### Example, asymptotic form of the Bessel function ${\displaystyle J_{0}(\lambda r)}$ for large ${\displaystyle \lambda r>0}$

We consider the integral representation of a Bessel function

${\displaystyle {\displaystyle J_{0}(\lambda r)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }e^{i\lambda r\sin(\theta )}\,d\theta .}}$

• ${\displaystyle \lambda r>0}$ is the large parameter. Here, both ${\displaystyle \lambda >0}$ and ${\displaystyle r>0}$
• ${\displaystyle \phi (\theta )=\sin(\theta )}$, ${\displaystyle \phi ^{\prime }(\theta )=\cos(\theta )}$, and ${\displaystyle \phi ^{\prime \prime }(\theta )=-\sin(\theta )}$
• the critical points are simple interior stationary points at ${\displaystyle \theta =\pm \pi /2}$
• we apply the stationary phase formula for simple interior stationary points to each stationary point and combine the results

${\displaystyle J_{0}(\lambda r)\sim {\frac {1}{2\pi }}{\sqrt {\frac {2\pi }{\lambda r}}}\left[e^{i\lambda r-i\pi /4}+e^{-i\lambda r+i\pi /4}\right]+O((\lambda r)^{-3/2})}$.

Simplifying, yields the common asymptotic form of the Bessel function for large ${\displaystyle \lambda r>0}$

${\displaystyle J_{0}(\lambda r)\sim {\sqrt {\frac {2}{\pi \lambda r}}}\cos(\lambda r-\pi /4)+O((\lambda r)^{-3/2})}$.

Thus, at the end of this analysis, we have the asymptotic approximation as well a an order estimate of the error. In this case, the product ${\displaystyle \lambda r}$ might further be reduced into a large ${\displaystyle r}$ approximation, or a large ${\displaystyle \lambda }$ approximation, where either the ${\displaystyle \lambda }$ or the ${\displaystyle r}$, respectively, become part of the phase.

# Multidimensional stationary phase

For problems involving Fourier-like integrals in higher dimensions

${\displaystyle I(\lambda )=\int _{D}f({\boldsymbol {x}})e^{i\lambda \phi ({\boldsymbol {x}})}\;d{\boldsymbol {x}}}$.

as ${\displaystyle \lambda \rightarrow \infty }$. Here, ${\displaystyle D}$ is a subdomain in ${\displaystyle n-}$dimensions, and ${\displaystyle d{\boldsymbol {x}}}$ is an ${\displaystyle n-}$dimensional hypervolume (or hypersurface element).

### The Multidimensional stationary phase formula

For an interior stationary point, the asymptotic representation of the multidimensional Fourier-like integral, for a simple stationary point, which is to say a point ${\displaystyle {\boldsymbol {x}}={\boldsymbol {x}}_{0}}$ where ${\displaystyle \nabla \phi ({\boldsymbol {x}}_{0})=0}$ but for which the determinant of the Hessian matrix ${\displaystyle \det A\equiv \det {\frac {\partial ^{2}\phi ({\boldsymbol {x}}_{0})}{\partial x^{\alpha }\partial x^{\beta }}}\neq 0}$. Here ${\displaystyle \alpha ,\beta =1,2,...,n}$. As with other discussions, the large parameter ${\displaystyle \lambda >0}$

${\displaystyle I(\lambda )\sim \left[{\frac {2\pi }{\lambda }}\right]^{n/2}{\frac {f({\boldsymbol {x}}_{0})}{\left.{\sqrt {|\det A|}}\right|_{{\boldsymbol {x}}={\boldsymbol {x}}_{0}}}}e^{i\lambda \phi ({\boldsymbol {x}}_{0})+{\frac {i\pi }{4}}{\mbox{sig}}(A)}+O(\lambda ^{-n})}$

as ${\displaystyle \lambda \rightarrow \infty }$. Here ${\displaystyle {\mbox{sig}}(A)}$ is the signature of the matrix ${\displaystyle A}$, which is the number of positive eigenvalues of ${\displaystyle A}$ minus the number of its negative eigenvalues.

## References

1. Bleistein, N. and Handelsman, R.A., (1986). Asymptotic expansions of integrals. Courier Corporation.
2. Bleistein, N. (1984). Mathematical methods for wave phenomena. Academic Press.
3. Erdélyi, A. (1956). Asymptotic expansions (No. 3). Courier Corporation.