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Estimating integrals
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[QUICK DESCRIPTION] Sometimes one can calculate a definite integral such as $\int_a^b f(x)\ dx$ (or more generally $\int_E f(x)\ d\mu(x)$, where $E$ is a measurable set, $f$ is an integrable function, and $\mu$ is a measure) exactly by antidifferentiating and using the fundamental theorem of calculus. However, sometimes this method does not work, or is not appropriate. (The second might be the case if, for example, one was integrating not an explicit function but an unspecified function about which one had certain information.) Also, in many applications one does not actually need to compute an integral exactly; merely obtaining a sufficiently good approximation to that integral, or even just an upper bound on its magnitude, may be sufficient in many applications (particularly those in analysis). This article gives links to further articles about techniques for approximating or bounding integrals. It is useful to make a distinction between ''non-oscillatory methods'', which do not try to exploit cancellation in the sign or phase of the integrand $f(x)$, and ''oscillatory methods'', which are designed to take advantage of such cancellation. For unsigned integrals, one should of course try the non-oscillatory methods; for signed integrals, the non-oscillatory methods tend to be easier but cruder (giving worse bounds), while the oscillatory methods give better bounds but are harder to implement. ===Links to articles=== * [[General methods for estimating integrals]] ** [[Divide and conquer]] *** [[Dyadic decomposition]] ** [[Treat an explicit but messy function abstractly]] ** [[Dimensional analysis]] * [[Methods for approximating integrals]] * [[Methods for bounding non-oscillatory integrals|Methods for bounding integrals (non-oscillatory)]]<comment thread="172" /> ** [[Bound the integrand by something simpler]] *** [[Which integrals are simpler to integrate]]? *** [[Base times height]] *** [[Bound by a Riemann sum]] ** [[Control level sets]] ** [[Replace a function by a "worst-case" variant]] *** [[Rearrangement inequalities]] *** [[Use convexity]] * [[Methods for bounding oscillatory integrals|Methods for bounding integrals (oscillatory)]] ** [[Use integration by parts to exploit cancellation]] *** [[The van der Corput lemma for oscillatory integrals]] ** [[The method of stationary phase]] ** [[Square and rearrange]] *** [[The van der Corput lemma for equidistribution]] ** [[Linearize the phase]] ** [[When controlling an oscillatory integral, bump functions and bounded phase corrections are not very important]] * [[Methods for simplifying integrals]] ** [[Which integrals are simpler to integrate]]? ** [[Use rescaling or translation to normalize parameters]] *** [[Select a favorable gauge]] ** [[Integration by parts]] ** [[Interchange integrals or sums]] ** [[Add and subtract something nearby that is simpler|Adding and subtracting]] ** [[Use Fourier identities]] ** [[Smoothing sums]] ** [[Shift the contour of integration]] * [[Methods for bounding integrals involving an arbitrary function $f(x)$]] ** [[w:Schur test]] ** [[Duality]] *** [[Holder's inequality]] *** The [[Cauchy-Schwarz inequality]]
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