Re: Integral of even functions


: Ken Pledger
In article <1180298502.925448.66080@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
esejdic@xxxxxxxxx wrote:

Hi there!
I came across an interesting thing. Integration of some even functions
from minus infinity to plus infinity is equal to zero. I have observed
similar thing for some Hermitian functions. Is this fact well known in
mathematics? If yes, what are the cases when this is true?
Thanks!


Just choose any function f on the positive reals whose integral
from 0 to infinity is zero, then define f(-x) = f(x) to extend it to
negative numbers as an even function. As a quite artificial example,
you could let f(x) =
1 for 0 < x < 3,
-1 for 3 <= x < 6,
0 for x >= 6.
It's pretty easy to construct as many such examples as you like.

Ken Pledger.

.

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