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 → ∞.

1 Reduction steps
2 One-dimensional case
3 See also
4 Reference

[edit] Reduction steps

The first major 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 + ….+ x_j² − x_{j + 1}² − x_{j + 2}² − … − x_n².

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 x_i. 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(x_i).

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.