# Terminology for $1/(e^x+1)$?

$\frac{1}{e^x+1}$ and $\frac{e^x}{e^x+1}$

Just wonder if either of the above function has a term/name associated with it? Or they are just functions that look beautiful without names? Maybe they appear very often under certain contexts?

I thought I might have seen it in some online courses. Maybe it was graphical model related or something else. But I'm not exactly sure right now and I cannot really find it on Google.

• If you change the plus signs to a minus sign and multiply by $x$, you have this. – alex.jordan Sep 10 '14 at 5:20
• the first is probably related to Fermi-Dirac distribution f(x) in physics. then second one is 1-f(x). – mike Sep 10 '14 at 5:20
• And Alex's variation appears in Planck's formula for black body radiation distribution. – Travis Sep 10 '14 at 5:21
• Both can be used in relation to the Fermi-Dirac distribution, with the second being used with certain probabilities of states. – Silynn Sep 10 '14 at 5:22
• Per Claude's comment, these functions also appear in the solutions to simple models of population growth in an environment with a fixed population capacity. – Travis Sep 10 '14 at 5:47

## 1 Answer

Hint

Quotic Wikipedia article :"A logistic function or logistic curve is a common special case of the more general sigmoid function, with equation $$f(x)=\frac{1}{1+e^{-x}}$$ So, multiplying numerator and denominator by $e^x$ $$f(x)=\frac{e^x}{1+e^{x}}$$ is just the same and $$g(x)=\frac{1}{1+e^{x}}=1-f(x)$$

• YEEEESSSSSS what I have seen was exactly a variant of the sigmoid function. Still don't remember where I saw it but this should be it! Thanks for all the answers! – StoneBird Sep 10 '14 at 6:09
• @StoneBird. You are very welcome ! – Claude Leibovici Sep 10 '14 at 6:22