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Concept
Stationary phase approximation
Summary
In mathematics, the stationary phase approximation is a basic principle of asymptotic analysis, applying to functions given by integration against a rapidly-varying complex exponential.
This method originates from the 19th century, and is due to George Gabriel Stokes and Lord Kelvin.
It is closely related to Laplace's method and the method of steepest descent, but Laplace's contribution precedes the others.
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.
Letting denote the set of critical points of the function (i.e. points where ), under the assumption that is either compactly supported or has exponential decay, and that all critical points are nondegenerate (i.e. for ) we have the following asymptotic formula, as :
Here denotes the Hessian of , and denotes the signature of the Hessian, i.e. the number of positive eigenvalues minus the number of negative eigenvalues.
For , this reduces to:
In this case the assumptions on reduce to all the critical points being non-degenerate.
This is just the Wick-rotated version of the formula for the method of steepest descent.
Consider a function
The phase term in this function, , is stationary when
or equivalently,
Solutions to this equation yield dominant frequencies for some and . If we expand as a Taylor series about and neglect terms of order higher than , we have
where denotes the second derivative of . When is relatively large, even a small difference 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,
This integrates to
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.
Official source
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