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.