Inmathematics, thestationary phase approximation is a basic principle ofasymptotic analysis, applying to functions given by integration against a rapidly-varying complex exponential.

This method originates from the 19th century, and is due toGeorge Gabriel Stokes andLord Kelvin.^[1]It is closely related toLaplace's method and themethod of steepest descent, but Laplace's contribution precedes the others.

The main idea of stationary phase methods relies on the cancellation ofsinusoids 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^{[clarification needed]}.

Letting${\displaystyle \mathrm{\Sigma }}$  denote the set ofcritical points of the function${\displaystyle f}$  (i.e. points where${\displaystyle \mathrm{\nabla }f=0}$ ), under the assumption that${\displaystyle g}$  is either compactly supported or has exponential decay, and that all critical points are nondegenerate (i.e.${\displaystyle det(\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}(f(x_0)))\ne 0}$  for${\displaystyle x_0\in \mathrm{\Sigma }}$ ) we have the following asymptotic formula, as${\displaystyle k\to \mathrm{\infty }}$ :

${\displaystyle {\int }_{{\mathbb{R}}^n}g(x)e^{ikf(x)}dx=\sum _{x_0\in \mathrm{\Sigma }}{e}^{ikf(x_0)}|det(\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}(f(x_0))){|}^{-1/2}{e}^{\frac{i\pi }{4}\mathrm{s}\mathrm{g}\mathrm{n}(\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}(f(x_0)))}(2\pi /k{)}^{n/2}g(x_0)+o(k^{-n/2})}$ 

Here${\displaystyle \mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}(f)}$  denotes theHessian of${\displaystyle f}$ , and${\displaystyle \mathrm{s}\mathrm{g}\mathrm{n}(\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}(f))}$  denotes thesignature of the Hessian, i.e. the number of positive eigenvalues minus the number of negative eigenvalues.

For${\displaystyle n=1}$ , this reduces to:

${\displaystyle {\int }_{\mathbb{R}}g(x)e^{ikf(x)}dx=\sum _{x_0\in \mathrm{\Sigma }}g(x_0)e^{ikf(x_0)+\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(f^{″}(x_0))i\pi /4}{\left(\frac{2\pi }{k|f^{″}(x_0)|}\right)}^{1/2}+o(k^{-1/2})}$ 

In this case the assumptions on${\displaystyle f}$  reduce to all the critical points being non-degenerate.

This is just theWick-rotated version of the formula for themethod of steepest descent.

Consider a function

${\displaystyle f(x,t)=\frac{1}{2\pi }{\int }_{\mathbb{R}}F(\omega )e^{i[k(\omega )x-\omega t]}d\omega }$ .

The phase term in this function,${\displaystyle \varphi =k(\omega )x-\omega t}$ , is stationary when

${\displaystyle \frac{d}{d\omega }\left(k(\omega )x-\omega t\right)=0}$ 

or equivalently,

${\displaystyle \frac{dk(\omega )}{d\omega }{|}_{\omega ={\omega }_0}=\frac{t}{x}}$ .

Solutions to this equation yield dominant frequencies${\displaystyle {\omega }_0}$  for some${\displaystyle x}$  and${\displaystyle t}$ . If we expand${\displaystyle \varphi }$  as aTaylor series about${\displaystyle {\omega }_0}$  and neglect terms of order higher than${\displaystyle (\omega -{\omega }_0{)}^2}$ , we have

${\displaystyle \varphi =\left[k({\omega }_0)x-{\omega }_{0}t\right]+\frac{1}{2}x{k}^{″}({\omega }_0)(\omega -{\omega }_0{)}^2+\cdots }$ 

where${\displaystyle k^{″}}$  denotes the second derivative of${\displaystyle k}$ . When${\displaystyle x}$  is relatively large, even a small difference${\displaystyle (\omega -{\omega }_0)}$  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,

${\displaystyle {\int }_{\mathbb{R}}{e}^{\frac{1}{2}ic{x}^2}dx=\sqrt{\frac{2i\pi }{c}}=\sqrt{\frac{2\pi }{|c|}}{e}^{\pm i\frac{\pi }{4}}}$ .

${\displaystyle f(x,t)\approx \frac{1}{2\pi }{e}^{i\left[k({\omega }_0)x-{\omega }_{0}t\right]}\left|F({\omega }_0)\right|{\int }_{\mathbb{R}}{e}^{\frac{1}{2}ix{k}^{″}({\omega }_0)(\omega -{\omega }_0{)}^2}d\omega }$ .

This integrates to

${\displaystyle f(x,t)\approx \frac{\left|F({\omega }_0)\right|}{2\pi }\sqrt{\frac{2\pi }{x\left|k^{″}({\omega }_0)\right|}}\cos \left[k({\omega }_0)x-{\omega }_{0}t\pm \frac{\pi }{4}\right]}$ .

The first major general statement of the principle involved is that the asymptotic behaviour ofI(k) depends only on thecritical points off. If by choice ofg the integral is localised to a region of space wheref has no critical point, the resulting integral tends to 0 as the frequency of oscillations is taken to infinity. See for exampleRiemann–Lebesgue lemma.

The second statement is that whenf is aMorse function, so that the singular points off arenon-degenerate and isolated, then the question can be reduced to the casen = 1. In fact, then, a choice ofg can be made to split the integral into cases with just one critical pointP in each. At that point, because theHessian determinant atP is by assumption not 0, theMorse lemma applies. By a change of co-ordinatesf may be replaced by

${\displaystyle (x_1^2+x_2^2+\cdots +x_j^2)-(x_{j+1}^2+x_{j+2}^2+\cdots +x_n^2)}$ .

The value ofj is given by thesignature of theHessian matrix off atP. As forg, the essential case is thatg is a product ofbump functions ofx_i. Assuming now without loss of generality thatP is the origin, take a smooth bump functionh with value 1 on the interval[−1, 1] and quickly tending to 0 outside it. Take

${\displaystyle g(x)=\prod _{i}h(x_i)}$ ,

thenFubini's theorem reducesI(k) to a product of integrals over the real line like

${\displaystyle J(k)=\int h(x)e^{ikf(x)}dx}$ 

withf(x) = ±x². The case with the minus sign is thecomplex conjugate of the case with the plus sign, so there is essentially one required asymptotic estimate.

In this way asymptotics can be found for oscillatory integrals for Morse functions. The degenerate case requires further techniques (see for exampleAiry function).

The essential statement is this one:

${\displaystyle {\int }_{-1}^1{e}^{ik{x}^2}dx=\sqrt{\frac{\pi }{k}}{e}^{i\pi /4}+\mathcal{O}\left(\frac{1}{k}\right)}$ .

In fact bycontour integration it can be shown that the main term on the right hand side of the equation is the value of the integral on the left hand side, extended over the range${\displaystyle [-\mathrm{\infty },\mathrm{\infty }]}$  (for a proof seeFresnel integral). Therefore it is the question of estimating away the integral over, say,${\displaystyle [1,\mathrm{\infty }]}$ .^[2]

This is the model for all one-dimensional integrals${\displaystyle I(k)}$  with${\displaystyle f}$  having a single non-degenerate critical point at which${\displaystyle f}$  hassecond derivative${\displaystyle >0}$ . In fact the model case has second derivative 2 at 0. In order to scale using${\displaystyle k}$ , observe that replacing${\displaystyle k}$  by${\displaystyle ck}$ where${\displaystyle c}$  is constant is the same as scaling${\displaystyle x}$  by${\displaystyle \sqrt{c}}$ . It follows that for general values of${\displaystyle f^{″}(0)>0}$ , the factor${\displaystyle \sqrt{\pi /k}}$  becomes

${\displaystyle \sqrt{\frac{2\pi }{k{f}^{″}(0)}}}$ .

For  one uses the complex conjugate formula, as mentioned before.

As can be seen from the formula, the stationary phase approximation is a first-order approximation of the asymptotic behavior of the integral. The lower-order terms can be understood as a sum of overFeynman diagrams with various weighting factors, for well behaved${\displaystyle f}$ .

• Common integrals in quantum field theory
• Laplace's method
• Method of steepest descent

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• Victor Guillemin and Shlomo Sternberg (1990),Geometric Asymptotics, (see Chapter 1).
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• Aki, Keiiti; & Richards, Paul G. (2002),Quantitative Seismology (2nd ed.), pp 255–256. University Science Books,ISBN 0-935702-96-2
• Wong, R. (2001),Asymptotic Approximations of Integrals, Classics in Applied Mathematics, Vol. 34. Corrected reprint of the 1989 original. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. xviii+543 pages,ISBN 0-89871-497-4.
• Dieudonné, J. (1980),Calcul Infinitésimal, Hermann, Paris
• Paris, Richard Bruce (2011),Hadamard Expansions and Hyperasymptotic Evaluation: An Extension of the Method of Steepest Descents, Cambridge University Press, ISBN 978-1-107-00258-6

• "Stationary phase, method of the",Encyclopedia of Mathematics,EMS Press, 2001 [1994]