The integral over all space of ∫ cos(x²) dx has no right to converge, and yet it does.pic.twitter.com/FfFJIIOR0u
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@LukeBouck Does this escape your intuition of convergence? Yours is much better than mine
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This convergence makes sense to me. The integral on a finite interval of cos(nx)->0 as n->infty. The x^2 causes rapid oscillations as x->infty that act like increasing n in the cos(nx) example. I think @CoryGriffithpls has a good explanation.
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One thing I'll add is I think if I were to integrate cos(f(x)) over R, I would want the tail behavior of 1/|f(x)| to behave like a convergent integral. I'll make a conjecture that if integral of 1/|f(x)| from 1 to infty is convergent, then integral cos(f(x)) over R converges.
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