Heaviside step function

The Heaviside step function is zero for negative inputs and one for positive inputs. Its value at zero depends on convention.

Dirac delta function: A generalized function concentrated at one point, with unit integral and zero values elsewhere. The step’s distributional derivative is a Dirac delta at its jump.
Heaviside cover-up method: A shortcut for finding coefficients in certain partial-fraction decompositions. Oliver Heaviside developed a distinct operational technique also bearing his name.
Oliver Heaviside: An English electrical engineer and mathematician who developed operational methods for electromagnetic theory. The function is named for Heaviside, who used it to model switched electrical signals.
Sign function: A function equal to −1 for negative inputs, 0 at zero, and 1 for positive inputs. It shares the step’s jump location but changes the levels on either side.
Distribution (mathematics): A generalized function defined through its action on smooth test functions. Distribution theory gives precise meaning to the step’s derivative and point value.
Causal system: A system whose output at any time depends only on present and past inputs. Multiplying by a step can make a signal vanish before its start time.
Operational calculus: A collection of methods that treat differentiation and integration as algebraic operations on functions. Heaviside used step functions within this approach to solve circuit and field equations.
Rectangular function: A function that is constant on a finite interval and zero outside it. A finite pulse can be built from the difference of two shifted steps.
Laplace transform: An integral transform that maps functions of time to functions of a complex variable. The step’s delayed forms become simple factors in Laplace-domain equations.
Switching function: A time-dependent function that indicates when a system changes between operating states. The step is the ideal mathematical model of a one-way switch.

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