What does ∫(/a) mean?
For example: ∫(/a) = ∫(/a)(/a) = 1/a * ∫(p)(p) = 1/a
From what I understand, the delta function - (/a) is being compressed by a scaling factor a. Thus the area under the curve is "shrunk" by a.
What I don't understand is why do we need to "(/a)" ? I've only understood d(t) as "a very small increment of the unit".
Thanks for your time. I hope my question was specific enough.
I don't agree at all that ∫(/a) = ∫(/a)(/a), or with the rest of your computation for that matter.
But I can explain what the d(t/a) means. It means that your small increments of time are weighed down by how much the function $t\mapsto t/a$ varies in each of them.
In general, you can replace $g'(t)dt$ by $d(g(t))$ in an integral, for any function $g$. You usually do that when you are preparing a change of variables.
Indeed if you can write the integrand as a function of $g(t)$ too, then you take your new variable $p$ to represent $g(t)$.
For instance (assume $a>0$ for clarity)
$$\int \delta(t/a) dt = a \int \delta(t/a) \dfrac 1a dt = a\int \delta(t/a) d (t/a) = a\int \delta(p) dp = a.$$
In the first equality I introduced a $1/a$ term by mutiplicating by $a$. In the second one I recognized that $\dfrac 1a$ is the derivative of $t/a$. In the third one I performed the change of variables $p=1/a$.