Posers and Puzzles
27 Nov 04
29 Nov 04
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.
30 Nov 04
Originally posted by iamatigerBy this defintion, there are uninteresting integers, because each of the (presumably finite) number of people who've lived has only thought of finitely many numbers.
A number has interested someone, at some time, if someone has thought of it. Hence the set of uninteresting integers are all those integers with values nobody has thought of.
Can anyone prove that this set is empty?