Stationary phase approximation

Stationary phase approximation

In mathematics, the stationary phase approximation is a basic principle of asymptotic analysis, applying to oscillatory integrals

I(k) = \int g(x) e^{ikf(x)}\,dx

taken over n-dimensional space Rn where i is the imaginary unit. Here f and g are real-valued smooth functions. The role of g is to ensure convergence; that is, g is a test function. The large real parameter k is considered in the limit as k → ∞.



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.

An example

Consider a function

f(x,t) = \frac{1}{2\pi} \int_{\mathbb{R}} F(\omega) e^{i(kx - \omega t)} d\omega

The phase term in this function, ϕ = kx − ωt is "stationary" when

\frac{d}{d\omega}\left(k\left(\omega\right)x - \omega t\right) \approx 0

or equivalently,

\frac{d\omega}{dk} \approx \frac{x}{t}

Solutions to this equation yield dominant frequencies ωdom(x,t) for a given x and t. If we expand ϕ in a Taylor series about ωdom and neglect terms of order higher than (ω − ωdom)2,

\phi \sim k(\omega_{dom})x - \omega_{dom} t + \frac{x}{2}\frac{d^2k}{d\omega^2}(\omega-\omega_{dom})^2

When x is relatively large, even a small difference ω − ωdom 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 double the real contribution from the positive frequencies of the transform to account for the negative frequencies,

f(x, t) = \frac{1}{2\pi} 2 \mbox{Re}\left\{ \exp\left[i\left[k(\omega_{dom})x-\omega_{dom}t\right]\right] \left|F(\omega_{dom})\right| \int_{\mathbb{R}}\exp\left[i\frac{x}{2}\frac{d^2k}{d\omega^2}(\omega-\omega_{dom})^2\right]d\omega\right\}

This integrates to

f(x, t) \sim \frac{\left|F(\omega_{dom})\right|}{\pi} \sqrt{ \frac{2\pi}{x\left|\frac{d^2k}{d\omega^2}\right|}} \cos\left[ k(\omega_{dom})x - \omega_{dom}t \pm \frac{\pi}{4}\right]

Reduction steps

The first major general statement of the principle involved is that the asymptotic behaviour of I(k) depends only on the critical points of f. If by choice of g the integral is localised to a region of space where f has no critical point, the resulting integral tends to 0 as the frequency of oscillations is taken to infinity. See for example Riemann-Lebesgue lemma.

The second statement is that when f is a Morse function, so that the singular points of f are non-degenerate and isolated, then the question can be reduced to the case n = 1. In fact, then, a choice of g can be made to split the integral into cases with just one critical point P in each. At that point, because the Hessian determinant at P is by assumption not 0, the Morse lemma applies. By a change of co-ordinates f may be replaced by

x12 + ….+ xj2xj + 12xj + 22 − … − xn2.

The value of j is given by the signature of the Hessian matrix of f at P. As for g, the essential case is that g is a product of bump functions of xi. Assuming now without loss of generality that P is the origin, take a smooth bump function h with value 1 on the interval [−1,1] and quickly tending to 0 outside it. Take

g(x) = Π h(xi).

Then Fubini's theorem reduces I(k) to a product of integrals over the real line like

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

with f(x) = x2 or −x2. The case with the minus sign is the complex 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 example Airy function.

One-dimensional case

The essential statement is this one:

\int_{-1}^1 \exp\left(ikx^2\right)\,dx
= \sqrt{\pi \over k} \,\exp\left({\pi i \over 4}\right)
+ O\left({1 \over k}\right).

In fact by contour 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 [−∞,∞]. Therefore it is the question of estimating away the integral over, say, [1,∞].[1]

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

\sqrt{2\pi \over kf''(0)}.

For f″(0) < 0 one uses the complex conjugate formula, as was mentioned before.

See also


  • Bleistein, N. and Handelsman, R. (1975), Asymptotic Expansions of Integrals, Dover, New York.
  • Victor Guillemin and Shlomo Sternberg (1990), Geometric Asymptotics, (see Chapter 1).
  • 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.


  1. ^ See for example Jean Dieudonné, Infinitesimal Calculus, p. 119.

External links

Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • Stationary phase — The term stationary phase may refer to * Stationary phase (biology), a phase in bacterial growth * Stationary phase (chemistry), a medium used in chromatography * stationary phase approximation in the evaluation of integrals in mathematics …   Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

  • Common integrals in quantum field theory — There are common integrals in quantum field theory that appear repeatedly.[1] These integrals are all variations and generalizations of gaussian integrals to the complex plane and to multiple dimensions. Other integrals can be approximated by… …   Wikipedia

  • BRST quantization — In theoretical physics, BRST quantization (where the BRST refers to Becchi, Rouet, Stora and Tyutin) is a relatively rigorous mathematical approach to quantizing a field theory with a gauge symmetry. Quantization rules in earlier QFT frameworks… …   Wikipedia

  • Michael Atiyah — Sir Michael Atiyah Born 22 April 1929 (1929 04 22) (age 82) …   Wikipedia

  • Time-frequency representation — A time frequency representation (TFR) is a view of a signal (taken to be a function of time) represented over both time and frequency. Time frequency analysis means analysis of a TFR. TFRs are often complex valued fields over time and frequency,… …   Wikipedia

  • Duistermaat–Heckman formula — In mathematics, the Duistermaat–Heckman formula, due to Duistermaat and Heckman (1982), states that the pushforward of the canonical (Liouville) measure on a symplectic manifold under the moment map is a piecewise polynomial measure.… …   Wikipedia

  • Chirp — Un chirp linéaire de amplitude constante. Sur les autres projets Wikimedia : « Chirp », sur le Wiktionnaire (dictionnaire universel) Un chirp (mot d origine anglaise signifiant « gazouillis ») est par définition …   Wikipédia en Français

  • Dephasing rate SP formula — The SP formula for the dephasing rate Γφ of a particle that moves in a fluctuating environment, unifies various results that have been obtained, notably in condensed matter physics with regard to the motion of electrons in a metal [1] [2] [3] [4] …   Wikipedia

  • analysis — /euh nal euh sis/, n., pl. analyses / seez /. 1. the separating of any material or abstract entity into its constituent elements (opposed to synthesis). 2. this process as a method of studying the nature of something or of determining its… …   Universalium

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”