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The Heaviside step function, or the unit step function, usually denoted by ''H'' (but sometimes ''u'' or ''θ''), is a discontinuous function whose value is zero for negative argument and one for positive argument. 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. The simplest definition of the Heaviside function is as the derivative of the ramp function: : The Heaviside function can also be defined as the integral of the Dirac delta function: ''H''′ = ''δ''. 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.) In operational calculus, useful answers seldom depend on which values is used for ''H''(0), since ''H'' is mostly used as a distribution. However, the choice may have some important consequences in functional analysis and game theory, where more general forms of continuity are considered. Some common choices can be seen below. ==Discrete form== An alternative form of the unit step, as a function of a discrete variable ''n'': : or using the half-maximum convention: : where ''n'' is an integer. Unlike the usual (not discrete) case, the definition of ''H''() 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: : where : is the discrete unit impulse function. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Heaviside step function」の詳細全文を読む スポンサード リンク
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