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== Analytic approximations ==
[[File:Step function approximation.png|alt=A set of functions that successively approach the step function|thumb|500x500px|<math>\frac{1}{2} + \frac{1}{2} \tanh(kx) = \frac{1}{1+e^{-2kx}}</math> approaches the step function as <math>k \rightarrow \infty</math>.]]
For a [[Smooth function|smooth]] approximation to the step function, one can use the [[logistic function]]
<math display="block">H(x) \approx \tfrac{1}{2} + \tfrac{1}{2}\tanh kx = \frac{1}{1+e^{-2kx}},</math>
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