How can I Numerically integrate a function multiplied with a dirac delta function?

for instance, integrate f=x.^2.*dirac(x), type function from -1 to 1.

Thanks

Answer 1

[Roger Stafford]

The integral of f(x) = x.^2.*dirac(x) w.r. x from -1 to +1 is zero. That is because the dirac(x) function is considered 0 at all points x except at x = 0, and at x = 0 its value is so large that the integral across the single point x = 0 is just 1. However in the case of your f(x) that one is multiplied by x^2 so that makes the integral zero. If you had a function g(x) = cos(x)*dirac(x), its integral from -1 to +1 would be just cos(0)*1 = 1.

Obviously this cannot be justified in the ordinary numerical sense with finite numbers and only has its basis in what mathematicians call “measure theory”. In particular, you cannot obtain the correct answer using numerical integration.

[Walter Roberson]

And if you must integrate it numerically then you are going to have to use one of the approximations to the dirac delta; https://en.wikipedia.org/wiki/Dirac_delta_function#Approximations_to_the_identity

[Roger Stafford]

@Mostafa A. Jouybari: In answer to your question concerning this "Answers" entry, you can find the following sentence, "A rigorous approach to regarding the Dirac delta function as a mathematical object in its own right requires measure theory or the theory of distributions" at the Wikipedia website

https://en.wikipedia.org/wiki/Dirac_delta_function .