The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a
step function, named after
Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently develope ...
(1850–1925), the value of which is
zero for negative arguments and
one
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. I ...
for positive arguments.
It is an example of the general class of
step functions, all of which can be represented as
linear combinations of translations of this one.
The function was originally developed in
operational calculus for the solution of
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s, where it represents a signal that switches on at a specified time and stays switched on indefinitely.
Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently develope ...
, who developed the operational calculus as a tool in the analysis of telegraphic communications, represented the function as .
The Heaviside function may be defined as:
* a
piecewise function
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Pi ...
:
* using the
Iverson bracket notation: