 Stationary phase approximation

In mathematics, the stationary phase approximation is a basic principle of asymptotic analysis, applying to oscillatory integrals
taken over ndimensional space R^{n} where i is the imaginary unit. Here f and g are realvalued 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
Basics
The main idea of stationary phase methods relies on the cancellation of sinusoids with rapidlyvarying 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
The phase term in this function, ϕ = kx − ωt is "stationary" when
or equivalently,
Solutions to this equation yield dominant frequencies ω_{dom}(x,t) for a given x and t. If we expand ϕ in a Taylor series about ω_{dom} and neglect terms of order higher than (ω − ω_{dom})^{2},
When x is relatively large, even a small difference ω − ω_{dom} 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,
This integrates to
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 RiemannLebesgue lemma.
The second statement is that when f is a Morse function, so that the singular points of f are nondegenerate 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 coordinates f may be replaced by
 x_{1}^{2} + ….+ x_{j}^{2} − x_{j + 1}^{2} − x_{j + 2}^{2} − … − x_{n}^{2}.
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
with f(x) = x^{2} or −x^{2}. 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.
Onedimensional case
The essential statement is this one:
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 onedimensional integrals I(k) with f having a single nondegenerate 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
For f″(0) < 0 one uses the complex conjugate formula, as was mentioned before.
See also
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 0935702962
 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 0898714974.
Notes
 ^ See for example Jean Dieudonné, Infinitesimal Calculus, p. 119.
External links
 Hazewinkel, Michiel, ed. (2001), "Stationary phase, method of the", Encyclopaedia of Mathematics, Springer, ISBN 9781556080104, http://eom.springer.de/S/s087270.htm
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