I'm trying to evaluate
$$ \int_{-\infty}^{\infty}dxf(x)\delta(g(x)) $$ Where $\delta$ is the dirac delta function, and $g(x)$ has zeros at $\{x=x_n, n = 1,...,N\}$. Here're some of my steps: $$ \int_{-\infty}^{\infty}dxf(x)\delta(g(x)) = \sum_{n = 1}^N\int_{x_n-\epsilon}^{x_n+\epsilon}dxf(x)\delta(0+g'(x_n)(x-x_n)+O(\epsilon^2)) \\= \sum_{n = 1}^N\int_{-\infty}^{\infty}dxf(x)\delta(g'(x_n)(x-x_n)) $$ I'm not exactly sure if these steps look correct. For the first equality, I Taylor expanded $g(x)$ around each of the zeros, but for the remaining terms, is it on the order of $\epsilon^2$? Is dropping the higher order terms necessary to change the integrand back to the improper integral (the next line)?
Also generally speaking, do we need to assume $f(x)$ and $g(x)$ to be real functions?
Thanks!