Hyperbolic Functions

Definition

The hyperbolic functions are defined using the right half of the unit hyperbola.

$\sinh (x)=\frac{e^x-e^{-x}}{2}=\frac{e^{2x}-1}{2e^x}=\frac{1-e^{-2x}}{2e^{-x}}$ $\cosh (x)=\frac{e^x+e^{-x}}{2}=\frac{e^{2x}+1}{2e^x}=\frac{1+e^{-2x}}{2e^{-x}}$ $\tanh (x)=\frac{\sinh (x)}{\cosh (x)}=\frac{e^x-e^{-x}}{e^x+e^{-x}}=\frac{e^{2x}-1}{e^{2x}+1}$ $\mathrm{csch}(x)=\frac{1}{\sinh (x)}=\frac{2}{e^x-e^{-x}}=\frac{2e^x}{e^{2x}-1}$ $\mathrm{sech}(x)=\frac{1}{\cosh (x)}=\frac{2}{e^x+e^{-x}}=\frac{2e^x}{e^{2x}+1}$ $\coth (x)=\frac{\cosh (x)}{\sinh (x)}=\frac{e^x+e^{-x}}{e^x-e^{-x}}=\frac{e^{2x}+1}{e^{2x}-1}$

Facts

The Hyperbolic Tangent Function is used as an activation function of neural network.