| <<< Note 1125.0 by PARITY::MOORADIAN >>>
-< proof >-
>> problem:
>> --------
>> show that
>> --
>> / /
>> -- d (ax-b)dx=1/lal
>> (limits of integration are (-infinity to +infinity)
>> /
>> where d = delta and dx is derivitive
>> lal= absolute value of a
By "delta", I assume that you refer to Dirac's delta distribution,
defined as that distribution (on the real line) that transforms a
function to its value at the origin.
A "proof" is done by treating delta as a nice (e.g., infinitely
differentiable function) and simply making the variable transformation
ax-b --> z; the result falls out that way (the absolute value of a
follows from that you have to change the direction of the integration
if a<0).
This technique can be made stringent by constructing some sequence of
infinitely differentiable functions that "converge to delta", i.e.,
such that the (convolution) integral of the function f and members of
the sequence converge to f(0). By using, e.g., dominated convergence
(Lebesgue's theorem on ...), you should be able to prove what you need
to.
A functional-analysis-based approach to a stringent proof can be found
in any reasonable book on the theory of distributions (e.g., Laurent
Schwartz's original book on the subject: "Theorie des distributions",
published in Paris in the late fifties or early sixties).
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