S-shaped Functions

The logit (logistic sigmoid) function:
\delta(x) = \frac{1}{1 + \exp(x)}
The tanh function:
\tanh(x) = 2\sigma(x) - 1
The probit function:
\Phi(x) = \int_{-\infty}^x \mathcal{N}(\theta;0,1) \;\mathrm{d}\theta
where
\Phi\left(\sqrt{\frac{\pi}{8}}x\right) \approx \sigma(x)
For more details about this approximation, look at Figure 4.9 of Pattern Recognition and Machine Learning.