Heaviside step function
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 function H (x) ≈ 1 2 + 1 2 tanh k x = 1 1 + e − 2 k x, {\displaystyle H(x)\approx {\tfrac {1}{2}}+{\tfrac {1}{2}}\tanh kx={\frac {1}{1+e^{-2kx}}},}

where a larger k corresponds to a sharper transition at x = 0. If we take H (0) = 1 / 2, equality holds in the limit: H (x) = lim k → ∞ 1 2 (1 + tanh k x) = lim k → ∞ 1 1 + e − 2 k x. {\displaystyle H(x)=\lim _{k\to \infty }{\tfrac {1}{2}}(1+\tanh kx)=\lim _{k\to \infty }{\frac {1}{1+e^{-2kx}}}.}

There are many other smooth, analytic approximations to the step function. Among the possibilities are: H (x) = lim k → ∞ (1 2 + 1 π arctan k x) H (x) = lim k → ∞ (1 2 + 1 2 erf k x) {\displaystyle {\begin{aligned}H(x)&=\lim _{k\to \infty }\left({\tfrac {1}{2}}+{\tfrac {1}{\pi }}\arctan kx\right)\\H(x)&=\lim _{k\to \infty }\left({\tfrac {1}{2}}+{\tfrac {1}{2}}\operatorname {erf} kx\right)\end{aligned}}}

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.)

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.
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