Talk:Stationary phase approximation

Physics Mid‑importance

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Mathematics Mid‑priority

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Error in rough idea

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.

If the phases are unrelated, then the sinusoids add incoherently, not destructively. Destructive interference requires coherence, i.e. definite phase relationship. —Keenan Pepper 18:44, 25 July 2007 (UTC)[reply]

The above wording may not be the best, but it is trying to capture the difference between constructive and destructive interference, respectively, of a set of sinusoids that are approximately in phase and a set whose phase slews rapidly as you sweep across the set. Incoherence requires a random element that is missing here.

Question about the example

How did the dependency of $F(\omega )$ change into $F(\omega _{dom})$ ? —Preceding unsigned comment added by 85.144.162.128 (talk) 20:19, 12 September 2007 (UTC)[reply]

To me it is not clear if the k in $kx-\omega t$ is the same k as the the k in the stationary phase approximation. Maybe we should either change the definition or the example? 80.101.82.58 (talk) 23:15, 1 May 2012 (UTC)[reply]

The whole article seems very muddled. I suggest a major rewrite. Shall we start by agreeing on some primary or secondary sources that we think exemplify the method? Billlion (talk) 17:57, 9 May 2013 (UTC)[reply]

Assessment comment

The comment(s) below were originally left at Talk:Stationary phase approximation/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

In the example, how did the dependency of

F(\omega )

change into

F(\omega _{dom})

?

Last edited at 20:14, 12 September 2007 (UTC). Substituted at 06:57, 30 April 2016 (UTC)

Truly terrible writing

The first mention of the functions f or g occurs in the first sentence of the section titled Formula, as follows:

"Letting $\Sigma$ denote the set of critical points of the function $f$ (i.e. points where $\nabla f=0$ ), under the assumption that $g$ is either compactly supported or has exponential decay, and that all critical points are nondegenerate (i.e. $\det(\mathrm {Hess} (f(x_{0})))\neq 0$ for $x_{0}\in \Sigma$ ) we have the following asymptotic formula, as $k\to \infty$ :

\int _{\mathbb {R} ^{n}}g(x)e^{ikf(x)}dx=\sum _{x_{0}\in \Sigma }e^{ikf(x_{0})}|\det({\mathrm {Hess} }(f(x_{0})))|^{-1/2}e^{{\frac {i\pi }{4}}\mathrm {sgn} (\mathrm {Hess} (f(x_{0})))}(2\pi /k)^{n/2}g(x_{0})+o(k^{-n/2})

."

It is essential that when a function is referred to in Wikipedia, readers do not need to guess what the domain and codomain of the function are, or what kind of function it is (continuous, differentiable, continuously differentiable, etc.).

I hope that someone with at least minimal writing skills who is familiar with this subject can fix this. 2601:200:C082:2EA0:E14D:BE82:EC0D:1CD5 (talk) 04:43, 15 May 2023 (UTC)[reply]