How do I use this proved result for the Dirac Delta function to evaluate the following integral involving a Dirac Delta function of a function?

Question

I just finished writing a proof that:

$int_(a)^(b) g(x)delta[f(x)]dx = sum_(i=1)^(N) g(x_i)/(|f'(x_i)|)$

(An explicit proof is shown here by user3728501.)

How do I use this result to evaluate:

$int_(0)^(oo) e^(-tx)delta(cos omegax)dx$

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Answer 1 · 2017-09-06T10:32:03

I would not use the result you have derived, instead I would form an explicit expression for $delta(f(x)) = delta(cos omega x)$ using:

$delta(f(x)) = sum_(i=0)^N (delta(x-x_i))/abs(f'(x_i))$

which only has a contribution when $f(x)=cosomegax=0$ , then you can perform a discrete summation to infinity in the latter integral

The roots are:

$cos omegax=0 => omegax = (2n+1)pi/2 = npi+pi/2 \ \ \ n in NN$

And:

$f'(x) = -omega sin omega x$

At at any given root:

$f'(x) = -omega sin (npi+pi/2)$
$\ \ \ \ \ \ \ \ \ = -omega {sin(npi)cos(pi/2)+cos(npi)sin(pi/2)}$
$\ \ \ \ \ \ \ \ \ = -omega cos(npi}$

And we require:

$abs(f'(x)) = abs(-omega cos(npi}) = omega$

So the sum for $delta(f(x))$ is:

$delta(f(x)) = sum_(i=0)^N (delta(x-(2i+1)pi/2))/omega$

And using this result in the latter integral it reduces the integral into a summation over all the roots. Does that help?

I presume you are attempoting to derive a laplace transform, in which standard Laplace theorems would probably provide a quicker derivation.