# Sigmoid function

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Logistic-curve.png

A sigmoid function is a mathematical function that produces a sigmoid curve — a curve having an "S" shape. Often, sigmoid function refers to the special case of the logistic function shown at right and defined by the formula:

[itex]P(t) = \frac{1}{1 + e^{-t}}[itex]

In general, a sigmoid function is real-valued and differentiable, having a non-negative or non-positive first derivative, one local minimum, and one local maximum.

Besides the logistic function, sigmoid functions include the ordinary arc-tangent, the hyperbolic tangent, and the error function.

## Sigmoid functions in neural networks

Sigmoid functions are often used in neural networks to introduce nonlinearity in the model and/or to make sure that certain signals remain within a specified range. A popular neural net element computes a linear combination of its input signals, and applies a bounded sigmoid function to the result; this model can be seen as a "smoothed" variant of the classical threshold neuron.

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