## FANDOM

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The Heaviside function, also known as the unit step function or Heaviside step function and usually denoted as $u_c$ , is a discontinuous function that

$u_c(x)=\begin{cases}0&xc\end{cases}$

The Heaviside function can be defined as the integral of the Dirac delta function.

$u_c(t)=\int\limits_{-\infty}^t\delta(s-c)ds$

The Heaviside function is often used in differential equations to model non-continuous events such as force in a driven harmonic oscillator or voltage in a circuit. If the Dirac delta function represents force being applied on an object, the Heaviside function will represent its momentum.

The value at $x=c$ is usually of little importance, although it is sometimes defined as $x=\tfrac12$ .

It has the Laplace transform

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

for $c>0$ .