Re: Integrating Odd Functions over all x in R
From: Ron Sperber (ronsperber_at_optonline.net)
Date: 02/17/05
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Date: Wed, 16 Feb 2005 20:15:46 -0500
Wild Gnu wrote:
> Is the improper integral of an odd function over the whole real line
> equal to 0 even if the integral from 0 to infinity is divergent? My
> textbook says:
>
> int(x^3 dx, -inf < x < inf) = lim_n->-inf(int(x^3 dx, n < x <= 0))
> + lim_n->inf(int(x^3 dx, 0 <= x < n))
>
> and since the first limit on the right side of the equation does not
> exist the integral on the left side of the equation is divergent.
>
> But what about this:
>
> int(x^3 dx, -inf < x < inf) = lim_n->inf(int(x^3 dx, -n <= x <= n))
> = lim_n->inf(0)=0
>
> since x^3 is odd int(x^3 dx, -a <= x <= a) = 0?
>
> -wg
>
That isn't the definition of the improper integral though. If you are
going to allow both endpoints to diverge simultaneously, why go from -n
to n? Why not
int(x^3 dx,-inf<x<inf)=lim_n->inf(int(x^3 dx, -n <= x
<=n+1))=lim_n->inf(int(x^3 dx, n<=x<=n+1) = lim_n->inf
((n+1)^3-n^3)/3=infinity? The integral from -n to n looks nice but is
pretty arbitrary. On the other hand breaking it up as integral from
-infinity to 0 + 0 to infinity isn't arbitrary since if both those
integrals converge you can replace 0 by any real #.
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