Preface

This section is devoted to an important class of (elementary) functions, called the hyperbolic functions.

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Glossary

Hyperbolic Functions

Two particular combinations of exponential functions appear frquently in many applications that it has been worth-while to use special symbols for those combinations. The hyperbolic sine of a variable x, written sinhx (or shx), is defined by
\[ \sinh x = \frac{1}{2}\, e^x - \frac{1}{2}\, e^{-x} ; \]
the hyperbolic cosine of x, written coshx (or chx). is defined by
\[ \cosh x = \frac{1}{2}\, e^x + \frac{1}{2}\, e^{-x} . \]
Mathematica has dedicated commands for these functions: Sinh[x] and Cosh[x]. Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert. The hyperbolic functions are related to regular trigonometric functions due to Euler's formulas:
\[ \sin x = \frac{1}{2{\bf j}}\, e^{{\bf j} \,x} - \frac{1}{2{\bf j}}\, e^{-{\bf j} \,x} , \qquad \cos x = \frac{1}{2}\, e^{{\bf j} \,x} + \frac{1}{2}\, e^{-{\bf j} \,x} , \quad {\bf j}^2 =-1. \]
Four more hyperbolic functions are
\begin{align*} \tanh x &= \frac{\sinh x}{\cosh x} , \qquad \sech x = \frac{1}{\cosh x} , \\ \csch x &= \frac{1}{\sinh x} , \qquad \coth x = \frac{\cosh x}{\sinh x} . \end{align*}
Plot[{Sinh[x], Cosh[x]}, {x, -1.9, 1.9}, PlotLabels -> Placed[{"sinh x", "cosh x"}, {Scaled[1], Before}]]
Plot[{Tanh[x], Coth[x]}, {x, -1.4, 1.4}, PlotLabels -> Placed[{"tanh x", "coth x"}, {Scaled[1], Before}]]
Plot[{Sech[x], Csch[x]}, {x, -1.4, 1.4}, PlotLabels -> Placed[{"sech x", "csch x"}, {Scaled[1], After}]]
From the definitions of hyperbolic functions, it follows
\[ \cosh^2 x - \sinh^2 x =1, \]
an identity similar to the well-known identity cos²x + sin²x = 1 in circular trigonometry. Derivative formulas are also similar:
\[ \frac{\text d}{{\text d}x}\,\sinh x = \cosh x , \qquad \frac{\text d}{{\text d}x}\,\cosh x = \sinh x. \]
From above graph, we see that coshx and sinhx satisfy the following relations
\begin{align*} \cosh \left( x+y \right) &= \cosh x\,\cosh y + \sinh x\,\sinh y, \qquad \cosh \left( x-y \right) = \cosh x\,\cosh y - \sinh x\,\sinh y, \\ \sinh \left( x+y \right) &= \sinh x\,\cosh y + \cosh x \,\sinh y , \qquad \sinh \left( x-y \right) = \sinh x\,\cosh y - \cosh x\,\sinh y, \\ \sinh^2 y &= \frac{1}{2} \left( \cosh 2y - 1 \right) , \qquad \cosh^2 y = \frac{1}{2} \left( \cosh 2y +1 \right) , \\ \cosh 2y &= \cosh^2 y + \sinh^2 y = 2\,\cosh^2 y -1 = 2\,\sinh^2 y +1 , \\ \sinh 2y &= 2\,\sinh y\,\cosh y , \\ \tanh \left( x+y \right) &= \frac{\tanh x + \tanh y}{1+ \tanh x\,\tanh y} . \end{align*}

 

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