Dirac Delta

Related Derivation of the Heaviside Step Function to Dirac Delta Function Relation with Distribution Definition

\delta(x) = \left{\begin{matrix} 0 & x \ne 0 \\ \infty & x = 0 \end{matrix}\right. \ \begin{bmatrix} \frac{1}{\[x]} \end{bmatrix}

If the argument $x$ is a length in m, $\text{[}\delta (x)]=m^{-1}$

$$\int \_{-\infty }^{\infty }\delta (x)=1$$

For every function $f(x)$ the product with delta is only nonzero at $x=1$

$$f(x)\delta (x)=f(0)\delta (x)$$

the integral of such product equals $f(0)$

$$\int \_{-\infty }^{\infty }f(x)\delta (x)=f(0)$$

Delta is even

$$\delta (-x)=\delta (x)$$

Identities apply

$$\delta (kx)=\frac{1}{|k|}\delta (x)$$

Generalization for Shifting the Peak

\delta(x) = \left{\begin{matrix} 0 & x \ne a \\ \infty & x = a \end{matrix}\right. $$f(x)\delta (x-a)=f(a)\delta (x-a)$$ $$\int \_{-\infty }^{\infty }f(x)\delta (x-a)=f(a)$$

Identities

• $\nabla \frac{1}{r^2}=4\pi {\delta }^3(r)$
• ${\nabla }^2\frac{1}{r}=-4\pi {\delta }^3(r)$