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Topic: dirac delta function is not a TRUE function??
Replies: 11   Last Post: Jul 19, 2011 11:08 PM

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Joshua Cranmer

Posts: 487
Registered: 8/20/08
Re: dirac delta function is not a TRUE function??
Posted: Jul 12, 2011 1:02 AM
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On 7/11/2011 9:35 PM, MBALOVER wrote:
> Thanks Gerry. But I am still confused. I think at x = 0, delta(x) =
> infinity. If so, y = -log(0) = + infinity too. Is y = - log(x) Not a
> true function?

The domain of the logarithm function is x > 0, so 0 is not in the domain
of y = - log x. Hence any formula that uses logarithm becomes
meaningless when its argument reaches that value, so, for example, the
integral from -1 to 1 of log x is meaningless.

In contrast, with the Dirac delta function we end up attempting to make
sense of what goes on at x = 0. The integral from -1 to 1 of this
function is, unlike the logarithm, well-defined: it is 1. Since it is
interesting for its unusual integral value, it is only used in the
contexts where its definition as a function is invalid, unlike other
classical functions.

Beware of bugs in the above code; I have only proved it correct, not
tried it. -- Donald E. Knuth

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