A common example of a sigmoid function is the logistic function shown in the first figure and defined by the formula:
Other standard sigmoid functions are given in the Examples section. In some fields, most notably in the context of artificial neural networks, the term "sigmoid function" is used as an alias for the logistic function.
Special cases of the sigmoid function include the Gompertz curve (used in modeling systems that saturate at large values of x) and the ogee curve (used in the spillway of some dams). Sigmoid functions have domain of all real numbers, with return (response) value commonly monotonically increasing but could be decreasing. Sigmoid functions most often show a return value (y axis) in the range 0 to 1. Another commonly used range is from −1 to 1.
A sigmoid function is a bounded, differentiable, real function that is defined for all real input values and has a non-negative derivative at each point and exactly one inflection point. A sigmoid "function" and a sigmoid "curve" refer to the same object.
Inverted logistic S-curve to model the relation between wheat yield and soil salinity. 
Many natural processes, such as those of complex system learning curves, exhibit a progression from small beginnings that accelerates and approaches a climax over time. When a specific mathematical model is lacking, a sigmoid function is often used.
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Mitchell, Tom M. (1997). Machine Learning. WCB–McGraw–Hill. ISBN 978-0-07-042807-2.. In particular see "Chapter 4: Artificial Neural Networks" (in particular pp. 96–97) where Mitchell uses the word "logistic function" and the "sigmoid function" synonymously – this function he also calls the "squashing function" – and the sigmoid (aka logistic) function is used to compress the outputs of the "neurons" in multi-layer neural nets.
Humphrys, Mark. "Continuous output, the sigmoid function". Properties of the sigmoid, including how it can shift along axes and how its domain may be transformed.