Integral with delta Dirac power

Question

Is it possible to calculate the integral: $$J=\int_{-\infty}^{+\infty}f(x)\delta(x-x_0)^kdx$$ wih $k\in\mathbb{R}$? I know that in the Colombeau algebra the distribution $\delta(x)^2$ is defined. What happens if the Delta function is raised to a real number different from $2$? Thanks in advance.

Answer 1

For $k$ being an integer:

You certainly know that

$\int f(x) \delta (x-x_0) dx = f(x_0)$

This is true regardless of what $f(x)$ is, even when $f(x)$ itself contains a $\delta$-function.

So your integral gives the highly singular result:

$\int f(x)\delta^{k-1}(x-x_0) \delta(x-x_0) dx = f(x_0)\delta^{k-1}(x_0-x_0)$

But you asked about all reals. Don't know what to tell you for non-integer $k$.