Integral involving Dirac delta function over a finite interval

Question

During the course of a textbook problem, I obtain the following (simplified to keep only important elements) :

$$\int^{b}_{-b}dy\int^{b}_{-b}dy' \space exp\{A(y^{2}-y'^{2})\} \space \delta(y-y')$$

where the answer is $2b$

My instinct would be to treat the problem as such :

$$\int^{b}_{-b}dy\int^{b}_{-b}dy' \space f(y, y') \space \delta(y-y')$$

where $\space f(y, y') = exp\{A(y^{2}-y'^{2})\}$

computing the first integral would yield

$$\int^{b}_{-b}dy \space f(y=y'|-b\leq y \leq b) = \int^{b}_{-b}dy = 2b$$

I understand the dirac distribution if very often misused and that bad shortcuts are often taken, I apologise for writing what is probably horrible mathematics.

My question however is whether the first double integral is well-defined, and whether the Dirac delta function can ever make any sense under an integral whose limits aren't infinity, for example in the case :

$$\int^{a}_{b}dy\int^{c}_{d}dy' \space f(y, y') \space \delta(y-y')$$

Answer 1

The double integral makes sense but its value depends on the intersection (if any) of the intervals $[b,a]$ and $[d,c]$. If they intersect on the interval $[f,e]$ then

$\int^{a}_{b}dy\int^{c}_{d}dy' \space f(y, y') \space \delta(y-y') = \int^{e}_{f} f(y,y) \space dy$

If they do not intersect then there is no point within the domain of the double integral at which $y=y'$ so

$\int^{a}_{b}dy\int^{c}_{d}dy' \space f(y, y') \space \delta(y-y') = 0$

In your example the two intervals are the same so

$\int^{b}_{-b}dy\int^{b}_{-b}dy' \space exp\{A(y^{2}-y'^{2})\} \space \delta(y-y') = \int^{b}_{-b} \space exp\{A(y^2-y^2)\} \space dy = \int^{b}_{-b} dy = 2b$

Answer 2

The Dirac distribution $\delta_x$ (the notation $\delta(x-y)$ is obtained by representing it as an integral kernel) is defined as the linear functional that maps a smooth function $f$ to the value $f(x)$. This definition does not really depend on the bounds of the domain of definition. Especially since in mathematical terms the Dirac distribution has as support the origin (or a translate thereof, i.e. any point $x$). So the behaviour of functions away of this point of support does not matter.

PS: I did not go to deep into the mathematical details, since only the application seemed to be relevant.