Scaling Property of Dirac Delta Function

Theorem
Let $\delta \left({t}\right)$ be the Dirac delta function.

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

Then:
 * $\delta \left({a t}\right) = \dfrac {\delta \left({t}\right)} {\left \vert a \right \vert}$

Proof
The equation can be rearranged as:
 * $\left\vert a \right\vert \delta \left({a t}\right) = \delta \left({t}\right)$

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

Definition of Dirac delta function:
 * $\left({1}\right):\delta \left({t}\right) = \begin{cases}

+\infty & : t = 0 \\ 0 & : \text{otherwise} \end{cases}$
 * $\left({2}\right):\displaystyle \int_{-\infty}^{+\infty} \delta \left({t}\right) \rd t = 1$

$\left({1}\right):$

$\left({2}\right):$

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, $\left\vert a \right\vert \delta \left({a t}\right) = \delta \left({t}\right)$.

The result follows after rearrangement.