seanieg89
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- 2007
Fun little exercise for those with some familiarity with convergence tests for integrals. (A first year university course in calculus should suffice).
\, dx$\\ converge? $)
Remarks:
Note that we only ever have "bad behaviour" at one of 0 or infinity, never both. Hence the notion of a Cauchy principal value is not needed to make sense of our integrals.
One moral of this computation is that a function can "grow very fast" and still be integrable over the real line if it oscillates "even faster", as the rapid oscillations lead to many cancellations. One can get a feel for this phenomenon by graphing the integrand for various values of alpha and beta.
Results of this sort can be generalised to much less nice functions than polynomials, and into higher dimensional space as well.
People who have done things with Fourier analysis before may notice a conceptual similarity with the Riemann-Lebesgue lemma, indeed the Fourier transform is pretty much the prototypical example of an oscillatory integral.
Remarks:
Note that we only ever have "bad behaviour" at one of 0 or infinity, never both. Hence the notion of a Cauchy principal value is not needed to make sense of our integrals.
One moral of this computation is that a function can "grow very fast" and still be integrable over the real line if it oscillates "even faster", as the rapid oscillations lead to many cancellations. One can get a feel for this phenomenon by graphing the integrand for various values of alpha and beta.
Results of this sort can be generalised to much less nice functions than polynomials, and into higher dimensional space as well.
People who have done things with Fourier analysis before may notice a conceptual similarity with the Riemann-Lebesgue lemma, indeed the Fourier transform is pretty much the prototypical example of an oscillatory integral.