Stationary phase approximation

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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 the i = √−1. 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 → ∞.

Contents

[edit] Basics

The main idea of stationary phase methods rely 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 destructively.

[edit] 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}(kx - \omega t) \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]

[edit] 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. 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, 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 + ….+ xj2 − xj + 12 − xj + 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.

[edit] 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 RHS is the value of the integral on the LHS, extended over the range [−∞,∞]. Therefore it is the question of estimating away the integral over, say, [1,∞].

(See for example Jean Dieudonné, Infinitesimal Calculus, p.119). 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{\pi \over kf''(0)}.

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

In this way asymptotics can be found for oscillatory integrals for Morse functions. The degenerate case requires further techniques. See for example Airy function.

[edit] See also

[edit] References

  • 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
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