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DRAFT: When is a function oscillatory?

By Richard Fateman


Numerical integration or “quadrature ” of an “oscillatory ” function f(x) is difficult in the sense that— if you don’t recognize that f(x) is oscillatory—some usually-good methods will be inaccurate, or fail to converge to a correct answer. On the other hand, if you can correctly determine that f(x) can be manipulated into one of several plausible oscillatory “templates ” say g(x) sin(mx) then there are special methods that are both fast and accurate. A quick and accurate categorization of some level of oscillation is therefore handy as an adjunct to numerical integration, especially as a potential preliminary step to symbolically re-arranging an integrand to suit an oscillatory method.

Year: 2009
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