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2.24 Hyperbolic functions

Hyperbolic functions are defined in terms of the exponential function by

cosh⁡x=ex+e-x2, sinh⁡x=ex-e-x2, tanh⁡x=sinh⁡xcosh⁡x=ex-e-xex+e-x,\cosh x={{e^{x}+e^{-x}}\over{2}},\quad\sinh x={{e^{x}-e^{-x}}\over{2}},\quad% \tanh x={{\sinh x}\over{\cosh x}}={{e^{x}-e^{-x}}\over{e^{x}+e^{-x}}},

and are called the hyperbolic cosine, hyperbolic sine and hyperbolic tangent functions respectively. The h stands for hyperbolic. The hyperbolic functions are useful in calculating integrals and solving differential equations, as in 3.36.

Catenary. Take a heavy chain and support its ends at two points at equal height, and let the chain hang freely under gravity. Then the shape of the suspended chain is a catenary, which is the graph of cosh\cosh. [The term catenary was introduced by Thomas Jefferson, third President of the USA.]