heaviside_and_dirac

Heaviside and Dirac Functions

Definition

Heaviside step function

The Heaviside (unit step) function $u_c(t)$ is zero before $t=c$ and one from $t=c$ onward:

$$u_c(t)=\{\begin{array}{ll}0& t<c,\\ 1& t\ge c.\end{array}$$

Dirac delta function

The Dirac delta function $\delta (t)$ is defined by the property

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

for any function $f(t)$. It is technically a distribution, not a function.

Intuition

The Heaviside function models an instantaneous switch: a system that is “off” before time $c$ and “on” after. The Dirac delta models an idealised instantaneous impulse — infinite magnitude, zero duration, unit area. Physically, $\delta (t)$ is the derivative of $u_0(t)$: a sudden jump in value corresponds to an infinite spike in its rate of change.

Formal Description

Heaviside properties

The Laplace transform is

$$\mathcal{L}\left\{u_c(t)\right\}=\frac{e^{-cs}}{s}.$$

The Heaviside function represents a time translation of $f(t)$ by $c$:

$$u_c(t)f(t-c)=\{\begin{array}{ll}0& t<c,\\ f(t-c)& t\ge c,\end{array}\mathcal{L}\left\{u_c(t)f(t-c)\right\}=e^{-cs}F(s).$$

A piecewise function $f(t)=f_1(t)$ for $t<c$ and $f_2(t)$ for $t\ge c$ can be written as

$$f(t)=f_1(t)+(f_2(t)-f_1(t))u_c(t).$$

Dirac delta properties

The shifted delta function can be represented as a limit of step functions:

$$\delta (t-c)=\underset{ϵ\to 0}{\lim }\frac{1}{2ϵ}\left(u_{c-ϵ}(t)-u_{c+ϵ}(t)\right).$$

Its Laplace transform (with $c>0$) is

$$\mathcal{L}\left\{\delta (t-c)\right\}=e^{-cs}.$$

Key Results

• The Heaviside function is the primitive (antiderivative) of the Dirac delta: $\frac{d}{dt}{u}_c(t)=\delta (t-c)$.
• Impulsive forcing via $\delta (t)$ produces a jump in the first derivative of the solution but not in the solution itself.
• Both functions appear in the Laplace transform table (entries 12–14).

Applications

• Encoding piecewise-defined forcing terms in ODE problems as a single expression.
• Modelling instantaneous forces (hammer blow, short circuit) in physics and engineering.
• Signal processing: the unit step and impulse are the canonical test inputs for linear time-invariant systems.

Trade-offs

• The Dirac delta is not a function in the classical sense; rigorous treatment requires the theory of distributions.
• Piecewise expressions using $u_c(t)$ can look compact but become unwieldy with many switch times.
• In numerical computation, approximating $\delta (t)$ requires care to avoid large discretisation errors.

Links

• Laplace Transform
• Inhomogeneous Second-Order ODE