Question

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?

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`

Answer 1

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.