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Wikipedia / Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), 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 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. Heaviside developed the operational calculus as a tool in the analysis of telegraphic communications and represented the function as .

Formulation

Taking the convention that , the Heaviside function may be defined as:

For the alternative convention that , it may be expressed as:

Other definitions that are undefined at include:

Relationship with Dirac delta

The Dirac delta function is the weak derivativeHence the Heaviside function can be considered to be the integralalthough this expansion may not hold (or even make sense) for , depending on which formalism one uses to give meaning to integrals involving . In this context, the Heaviside function is the cumulative distribution function of a random variable that is almost surely 0. (See Constant random variable.)

Analytic approximations

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.

For a smooth approximation to the step function, one can use the logistic functionwhere a larger corresponds to a sharper transition at .

If we take

There are many other smooth, analytic approximations to the step function.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.)

One could also use a scaled and shifted Sigmoid function.

In general, any cumulative distribution function of a continuous probability distribution that is peaked around zero and has a parameter that controls for variance can serve as an approximation, in the limit as the variance approaches zero. For example, all three of the above approximations are cumulative distribution functions of common probability distributions: the logistic, Cauchy and normal distributions, respectively.

Non-analytic approximations

Approximations to the Heaviside step function could be made through Smooth transition function

Integral representations

Often an integralwhere 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 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 . Indeed when is considered as a distribution or an element of (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.

Discrete form

(that is, taking in a discrete variable Or using the half-maximum convention:where is an integer. If is an integer, then must imply that , while must imply that the function attains unity at . Therefore the "step function" exhibits ramp-like behavior over the domain of , and cannot authentically be a step function, using the half-maximum convention.

Unlike the continuous case, the definition of is significant.

This function is the cumulative summation of the Kronecker delta is the discrete unit impulse function.

Antiderivative and derivative

The ramp function is an antiderivativeThe distributional derivative of the Heaviside step function is the Dirac delta function

Fourier transform

The Fourier transformHere is the distribution that takes a test function to the Cauchy principal value. The limit appearing in the integral is also taken in the sense of (tempered) distributions.

Unilateral Laplace transform

The Laplace transform of the Heaviside step function is a meromorphic function

When the bilateral transform is used, the integral can be split in two parts and the result will be the same.

See also

References