Definition:Heaviside Step Function
Definition
Let $c \ge 0$ be a constant real number.
The Heaviside step function on $c$ is the real function $u_c: \R \to \R$ defined as:
- $\map {u_c} t := \begin{cases} 1 & : t > c \\ 0 & : t < c \end{cases}$
If $c = 0$, the subscript is often omitted:
- $\map u t := \begin{cases} 1 & : t > 0 \\ 0 & : t < 0 \end{cases}$
There is no universal convention for the value of $\map {u_c} c$.
However, since $u_c$ is piecewise continuous, the value of $u_c$ at $c$ is usually irrelevant.
Heaviside Step Function with Two Variables
Let $u: \R \to \R$ be the Heaviside step function.
Then the Heaviside step function of two variables is the real function $u : \R^2 \to \R$ defined as the product of two step functions of one variable:
- $\map u {x, y} := \map u x \map u y$
In other words:
- $\map u {x, y} := \begin{cases} 1 & : \paren {x > 0} \land \paren {y > 0} \\ 0 & : \text {otherwise} \end{cases}$
Graph of Heaviside Step Function
The graph of the Heaviside step function is illustrated below:
Off and On
Let $t$ be understood as time.
Let be $f$ a function of $t$ used to model some physical process.
Off
- $\map {u_c} t \map f t$
can be understood as:
- $f$ is off until time $c$.
or:
- $f$ does not start until time $c$.
On
- $\map {u_c} t \map f t$
can be understood as:
- $f$ is on after time $c$.
Also denoted as
The Heaviside step function can also be denoted:
- $\map {H_c} t$
- $\map {\theta_c} t$
Variants of the letter $u$ can be found:
- $\map {\UU_c} t$
- $\map {\operatorname u_c} t$
Some sources bypass the need to use a subscript, and present it as:
- $\map {\UU} {t - c} = \begin {cases} 1 & : t > c \\ 0 & : t < c \end {cases}$
Also known as
This is also called the unit step function.
Some sources merge the terminology and refer to it as Heaviside's unit function, or Heaviside's unit step.
Also see
- Results about the Heaviside step function can be found here.
Source of Name
This entry was named for Oliver Heaviside.
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Special Functions: $\text {VII}$. The Unit Step function
- 1978: Ronald N. Bracewell: The Fourier Transform and its Applications (2nd ed.) ... (previous) ... (next): Frontispiece
- 1978: Ronald N. Bracewell: The Fourier Transform and its Applications (2nd ed.) ... (previous) ... (next): Chapter $4$: Notation for some useful Functions: Summary of special symbols: Table $4.1$ Special symbols
- 1978: Ronald N. Bracewell: The Fourier Transform and its Applications (2nd ed.) ... (previous) ... (next): Inside Back Cover
- 2009: William E. Boyce and Richard C. DiPrima: Elementary Differential Equations and Boundary Value Problems (9th ed.): $\S 6.3$
- For a video presentation of the contents of this page, visit the Khan Academy.