Heaviside step function

The Heaviside step function, using the half-maximum convention

The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument:

<math>u(x)=\begin{cases} 0, & x < 0 \\ 1, & x > 0 \end{cases}</math>

The function is used in the mathematics of control theory and signal processing to represent a signal that switches on at a specified time and stays switched on indefinitely.

It is the cumulative distribution function of a random variable which is almost surely 0. (See constant random variable.)

The Heaviside function is the integral of the Dirac delta function.

<math> u(x) = \int_{-\infty}^x { \delta(t)} dt </math>

The value of u(0) is occasionally of disputed value. Some writers give u(0) = 0, some u(0) = 1. u(0) = 1/2 is the most consistent choice used, since it maximizes the symmetry of the function and becomes completely consistent with the signum function. This makes for a more general definition:

<math> u(x) =

 \begin{cases} 0,           & x < 0
            \\ \frac{1}{2}, & x = 0
            \\ 1,           & x > 0
 \end{cases}

</math>

<math> u(x) = \frac{1}{2} \left ( 1 + \sgn(x) \right ) </math>

To remove the ambiguity of which value to use for u(0), a subscript specifying which value may be used:

<math> u_n(x) =

 \begin{cases} 0, & x < 0
            \\ n, & x = 0
            \\ 1, & x > 0
 \end{cases}

</math>

Often an integral representation of the step function is useful:

<math>u(x)=\lim_{ \epsilon \to 0} -{1\over 2\pi i}\int_{-\infty}^\infty {1 \over \tau+i\epsilon} e^{-i x \tau} d\tau </math>

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Heaviside step function

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