Review of the Hyperbolic Functions

Minkowski space is called a hyperbolic geometry. Clearly the time coordinate is not treated the same as the space coordinates. We will find that when space and time coordinates are mixed in a transformation, that we use the hyperbolic functions \bgroup\color{black}$ \sinh$\egroup and \bgroup\color{black}$ \cosh$\egroup instead of \bgroup\color{black}$ \sin$\egroup and \bgroup\color{black}$ \cos$\egroup. They are very symmetrically defined.

$\displaystyle e^{i\phi}$ $\displaystyle =\cos\phi+i\sin\phi$ $\displaystyle e^\phi$ $\displaystyle =\cosh\phi+\sinh\phi$    
$\displaystyle e^{-i\phi}$ $\displaystyle =\cos\phi-i\sin\phi$ $\displaystyle e^{-\phi}$ $\displaystyle =\cosh\phi-\sinh\phi$    
$\displaystyle \cos\phi$ $\displaystyle ={e^{i\phi}+e^{-i\phi}\over 2}$ $\displaystyle \cosh\phi$ $\displaystyle ={e^{\phi}+e^{-\phi}\over 2}$    
$\displaystyle \sin\phi$ $\displaystyle ={e^{i\phi}-e^{-i\phi}\over 2i}$ $\displaystyle \sinh\phi$ $\displaystyle ={e^{\phi}-e^{-\phi}\over 2}$    
$\displaystyle \tan\phi$ $\displaystyle ={e^{i\phi}-e^{-i\phi}\over e^{i\phi}+e^{-i\phi}}$ $\displaystyle \tanh\phi$ $\displaystyle ={e^{\phi}-e^{-\phi}\over e^{\phi}+e^{-\phi}}$    

Jim Branson 2012-10-21