29 Nov '04 17:25>
Originally posted by AcolyteUsually, the integral over R is defined as the limit as m and n go to infinity of the integral over [-m,n], provided this limit exists.
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That makes more sense.
From an analytical point-of-view it's a bit arbitrary to talk about the integral over R being the limit of the integral over [-n,n], as there's nothing special about 0, and an odd function g(x) is no more special than a translation of it, g(x-c), which won't be odd in general.
Given that the previous bit makes more sense, that also makes sense.
Nevertheless, the limit you defined is given a name, the Cauchy principle value of the integral, because it's useful some contexts, eg in conjunction with Cauchy's theorem, where the line -infinity to infinity is part of the limit of some loop in C you're integrating round, and the way you form the loop allows you to take the limit symmetrically about one of the axes.
I think I've seen this in the proof of the prime number theorem, but I only had a general notion of what was going on.
Thanks for the information.