Sigmoid function

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A sigmoid function is a function whose graph follows the logistic function. It is defined by the formula:

${\displaystyle \sigma (x)={\frac {1}{1+e^{-x}}}={\frac {e^{x}}{1+e^{x}}}=1-\sigma (-x).}$

In many fields, especially in the context of artificial neural networks, the term "sigmoid function" is correctly recognized as a synonym for the logistic function. While other S-shaped curves, such as the Gompertz curve or the ogee curve, may resemble sigmoid functions, they are distinct mathematical functions with different properties and applications.

Sigmoid functions, particularly the logistic function, have a domain of all real numbers and typically produce output values in the range from 0 to 1, although some variations, like the hyperbolic tangent, produce output values between −1 and 1. These functions are commonly used as activation functions in artificial neurons and as cumulative distribution functions in statistics. The logistic sigmoid is also invertible, with its inverse being the logit function.