Heaviside step function

• A piecewise function: 0 \\\\ 0, & x < 0 \\end{cases}"}}'>H(x):={1,x>00,x<0
• The derivative of the ramp function: δ(x)=ddx H(x),Hence the Heaviside function can be considered to be the integral of the Dirac delta function. This is sometimes written as:δ. 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

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  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 function:x = 0.

  If we take H(x)=limk→∞12(1+tanh⁡kx)=limk→∞11+e−2kx.

  H(x)=limk→∞(12+1πarctan⁡kx)H(x)=limk→∞(12+12erf⁡kx)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

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  Approximations to the Heaviside step function could be made through Smooth transition function like H(x)=limε→0+−12πi∫−∞∞1τ+iεe−ixτdτ=limε→0+ 12πi∫−∞∞1τ−iεeixτdτ,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

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  Since H is considered as a distribution or an element of H(0) = ⁠1/2⁠ is then an odd function. In this case the following relation with the sign function holds for all ∀x, H(x)+H(−x)=1.

• H is the indicator function of a closed semi-infinite interval: H needs to be left-continuous. In this case H(0) = [0,1].

An alternative form of the unit step, defined instead as a function H[n]={0,n<0,1,n≥0,Or using the half-maximum convention: 0, \\end{cases} "}}'>H[n]={0,n<0,12,n=0,1,n>0,where n < 0 must imply that n = 1. Therefore the "step function" exhibits ramp-like behavior over the domain of δ[n]=H[n]−H[n−1].This function is the cumulative summation of the Kronecker delta:∫−∞xH(ξ)dξ=xH(x)=max⁡{0,x}.The distributional derivative of the Heaviside step function is the Dirac delta function:p.v.∫−∞∞φ(s)sds. The limit appearing in the integral is also taken in the sense of (tempered) distributions.

The Laplace transform of the Heaviside step function is a meromorphic function. Using the unilateral Laplace transform we have: Weisstein, Eric W. "Heaviside Step Function". MathWorld.