The Heaviside step function, using the half-maximum convention
General information
General definition
Fields of application
Operational calculus
The Heaviside step function, or the unit step function, usually denoted by H or θ (but sometimes u, 1 or 𝟙), is a
step function named after
Oliver Heaviside, the value of which is
zero for negative arguments and
one for positive arguments. Different conventions concerning the value H(0) are in use. 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 equations, where it represents a signal that switches on at a specified time and stays switched on indefinitely.
Oliver Heaviside, who developed the operational calculus as a tool in the analysis of telegraphic communications, represented the function as 1.
Taking the convention that H(0) = 0, the Heaviside function may be defined as:
The
Dirac delta function is the
derivative of the Heaviside function:
Hence the Heaviside function can be considered to be the
integral of the Dirac delta function. This is sometimes written as
although this expansion may not hold (or even make sense) for x = 0, depending on which formalism one uses to give meaning to integrals involving δ. In this context, the Heaviside function is the
cumulative distribution function of a
random variable which is
almost surely 0. (See
Constant random variable.)
Approximations to the Heaviside step function are of use in
biochemistry and
neuroscience, where
logistic approximations of step functions (such as the
Hill and the
Michaelis–Menten equations) may be used to approximate binary cellular switches in response to chemical signals.
These limits hold
pointwise and in the sense of
distributions. In general, however, pointwise convergence need not imply distributional convergence, and vice versa distributional convergence need not imply pointwise convergence. (However, if all members of a pointwise convergent sequence of functions are uniformly bounded by some "nice" function, then
convergence holds in the sense of distributions too.)
Often an
integral representation of the Heaviside step function is useful:
where the second representation is easy to deduce from the first, given that the step function is real and thus is its own complex conjugate.
Zero argument
Since H is usually used in integration, and the value of a function at a single point does not affect its integral, it rarely matters what particular value is chosen of H(0). Indeed when H is considered as a
distribution or an element of L∞ (see
Lp space) it does not even make sense to talk of a value at zero, since such objects are only defined
almost everywhere. If using some analytic approximation (as in the
examples above) then often whatever happens to be the relevant limit at zero is used.
There exist various reasons for choosing a particular value.
H(0) = 1/2 is often used since the
graph then has rotational symmetry; put another way, H − 1/2 is then an
odd function. In this case the following relation with the
sign function holds for all x:
H(0) = 0 is used when H needs to be
left-continuous. In this case H is an indicator function of an
open semi-infinite interval:
In functional-analysis contexts from optimization and game theory, it is often useful to define the Heaviside function as a
set-valued function to preserve the continuity of the limiting functions and ensure the existence of certain solutions. In these cases, the Heaviside function returns a whole interval of possible solutions, H(0) = [0,1].
Discrete form
An alternative form of the unit step, defined instead as a function (that is, taking in a discrete variable n), is:
where n is an
integer. If n is an integer, then n < 0 must imply that n ≤ −1, while n > 0 must imply that the function attains unity at n = 1. Therefore the "step function" exhibits ramp-like behavior over the domain of [−1, 1], and cannot authentically be a step function, using the half-maximum convention.
Unlike the continuous case, the definition of H[0] is significant.
The discrete-time unit impulse is the first difference of the discrete-time step
This function is the cumulative summation of the
Kronecker delta:
The
Fourier transform of the Heaviside step function is a distribution. Using one choice of constants for the definition of the Fourier transform we have
Here p.v.1/s is the
distribution that takes a test function φ to the
Cauchy principal value of . The limit appearing in the integral is also taken in the sense of (tempered) distributions.