Dirac Delta Function

Consider the sequence of functions $\delta_n$ defined as:

$$\delta_n{(x)} = \begin{cases} n & \text{if } -\frac{1}{2n} \le x \le \frac{1}{2n} \\ 0 & \text{otherwise} \end{cases}$$

The Dirac delta function is the limit of $\delta_n$ as $n$ approaches infinity.

$$\delta = \lim_{n \to \infty} \delta_n(x) = \begin{cases} \infty & \text{if } x = 0 \\ 0 & \text{otherwise} \end{cases}$$

Properties

$$\int_{-\infty}^{\infty} \delta(x) \, dx = 1$$ $$\int_{-\infty}^{\infty} f(x)\delta(x-a) \, dx = f(a)$$ $$\mathcal{L}\{\delta(t)\} = 1$$ $$\mathcal{L}\{\delta(t-a)\} = e^{-as}$$