Dirac Delta Function definition with ksi (ξ)

Question

The dirac delta function has a definition

$$f(0)=\int_∞^∞f(x)δ(x)dx$$

and

$$ f(x)=\int_∞^∞f(x-ξ)δ(ξ)dξ $$

(the lower bound is minus infinity but I couldn't add a minus :/)

I do understand the first definition but I don't understand the second one. What is the idea behind the second notation with the ksi's?

I would appreciate it if someone could explain this/give the name of this type of notation so I can search a little further.

Thanks!!!

Answer 1

Integrating $f(x)$ against $\delta(x)$ gives $f(0)$, that is, evaluates $f$ at $x=0$.

In the second integral, let $g(\xi)=f(x-\xi)$, where $\xi$ is the variable and $x$ a parameter. Integrating $g(\xi)$ against $\delta(\xi)$ gives $g(0)=f(x)$.