Definition:Heaviside Step Function

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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.png


Also denoted as

  • $\map {H_c} t$
  • $\map {\theta_c} t$

Variants of the letter $u$ can be found:

  • $\map {\mathcal U_c} t$
  • $\map {\operatorname u_c} t$


Some sources bypass the need to use a subscript, and present it as:

$\map {\mathcal U} {t - c} = \begin{cases} 1 & : t > c \\ 0 & : t < c \end{cases}$


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.


Comment

If $t$ is understood as time and $f$ some function of time used to model some physical process, then:

$\map {u_c} t \, \map f t$

is often understood as:

$f$ is off until time $c$ and then on after time $c$

or:

$f$ does not start until time $c$.


Also see

  • Results about the Heaviside step function can be found here.


Source of Name

This entry was named for Oliver Heaviside.


Sources