Heaviside function

The Heaviside function, often written as H(x), is a non-continuous function whose value is zero for a negative input and one for a positive input.

The function is used in the mathematics of control theory to represent a signal that switches on at a specified time, and which stays switched on indefinitely. It was named after the Englishman Oliver Heaviside.

The Heaviside function is the integral of the Dirac delta function: H′(x) = δ(x). This is sometimes written as

 Discrete form 
We can also define an alternative form of the Heaviside step function as a function of a discrete variable n:

where n is an integer.

Or

The discrete-time unit impulse is the first difference of the discrete-time step

This function is the cumulative summation of the Kronecker delta:

where

is the discrete unit impulse function.

 Representations 
Often an integral representation of the Heaviside step function is useful:

 H(0) 
The value of the function at 0 can be defined as H(0) = 0, H(0) = ½ or H(0) = 1. In particular: 

Related pages
 Laplace transform
 Step function

 References 

Category:Mathematics