Delta function

i.e. Dirac delta function, unit impulse

$\delta(x)=\infty$, if $x=0$, else (when $x \neq 0$), $\delta(x)=0$. $\int_{-\infty}^{\infty} \delta(x) \, dx = 1$

a fundamental property of the delta function:

$$\int_{-\infty}^\infty f(x) \delta(x-a)\, dx = f(a)$$ and for $\epsilon > 0$, $$\int_{a-\epsilon}^{a+\epsilon} f(x) \delta(x-a)\, dx = f(a)$$

additional identities (for $x \neq a$):

$$\begin{align} \delta(x-a)=0 \\ \delta(a \, x) = \frac{1}{|a|} \delta(x) \\ \delta(x^2 - a^2) = \frac{1}{2|a|}[\delta(x+a) + \delta(x-a)] \end{align}$$

one of the Singularity functions, $$\mu_0(t) \triangleq \delta(t)$$ with impulse/Dirac delta $$\int_{-\infty}^t f(\lambda) \delta(\lambda - \tau)d\lambda = \begin{cases}0 \quad \text{for } t < \tau \\ f(\tau) \quad \text{for } t > \tau \end{cases}$$

Sifting property:

Discrete form: Kronecker delta

Derivative of Heaviside step function

References:

1. P. A. M. Dirac, The Principles of Quantum Mechanics. Oxford University Press, 1930.
2. https://en.wikipedia.org/wiki/Dirac_delta_function
3. https://mathworld.wolfram.com/DeltaFunction.html
4. https://lpsa.swarthmore.edu/BackGround/ImpulseFunc/ImpFunc.html