Scaling Property of Dirac Delta Function

Theorem
Let $\map \delta t$ be the Dirac delta function.

Let $a$ be a non zero constant real number.

Then:
 * $\map \delta {a t} = \dfrac {\map \delta t} {\size a}$

Proof
The equation can be rearranged as:
 * $\size a \map \delta {a t} = \map \delta t$

We will check the definition of Dirac delta function in turn.

Definition of Dirac delta function:
 * $\paren 1:\map \delta t = \begin{cases}

+\infty & : t = 0 \\ 0 & : \text{otherwise} \end{cases}$
 * $\paren 2:\ds \int_{-\infty}^{+\infty} \map \delta t \rd t = 1$

$\paren 1:$

$\paren 2:$

The proof of this part will be split into two parts, one for positive $a$ and one for negative $a$.

For $a > 0$:

For $a < 0$:

Therefore, by definition, $\size a \map \delta {a t} = \map \delta t$.

The result follows after rearrangement.