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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{Hyperbolic functions}


\subsubsection{Basic definitions, hyperbolic trigonometry}

\textit{Hyperbolic functions} sound like they ought to be some great new idea,
but they aren't. There are certain rules that they satisfy similarly to rules
for trig functions (with a twist), but it's not really a big deal. So they satisfy
identities similar to trig functions? That doesn't make them as important as
trig functions.  

However, you \textit{will} see these things mentioned from time to time, so
we so have to get the definitions straight. 

\begin{defn}
The \textit{hyperbolic sine} function is the function \( \sinh (x) \), which
is expressed as: 
\[
\sinh (x)\, :=\frac{1}{2}(e^{x}-e^{-x}).\]
 That's it. The \textit{hyperbolic cosine} function is the function \( \cosh x \)
which is expressed as: 
\[
\cosh (x)\, :=\frac{1}{2}(e^{x}+e^{-x}).\]

\end{defn}
The domain of either function is the whole real line, but the range is a bit
more subtle to see. For \( x \) large, \( \sinh (x) \) will also be large,
and for \( x \) ``large and negative'' (\( -x \) is then ``large''), \( \sinh (x) \)
will also be large in absolute value, but negative. So, since \( \sinh (x) \)
is continuous, the range of \( \sinh (x) \) is the whole real line. On the
other hand, since \( \cosh (x)=(e^{x}+e^{-x})/2, \) it is never negative, and
in fact is smallest when \( x=0 \) (which we can verify with the techniques
of Chapter 4 {[}Link{]}), \( \cosh (x)\geq 1 \), and in fact the range is \( \left\{ y\left| y\geq 1\right. \right\}  \). 

Now, the rest of the hyperbolic functions are defined as in real trigonometry:
\( \tanh (x):=\sinh (x)/\cosh (x) \), \( sech(x):=1/\cosh (x) \), \textit{etc.}

There are formulas similar to trig functions that do hold: 
\begin{eqnarray*}
\cosh ^{2}(x)-\sinh ^{2}(x) & = & 1,\\
\cosh (x+y) & = & \cosh (x)\, \cosh (y)+\sinh (x)\, \sinh (y),\\
\sinh (x+y) & = & \sinh (x)\, \cosh (y)+\cosh (x)\, \sinh (y).
\end{eqnarray*}
 There are, as with standard trigonometry, a large number of identities. Most
of them are remarkably similar to trigonometric identities, except, as with
these, for some changes in sign.

If this isn't enough, there are inverses, too. However, from the point of view
of calculus, even the trigonometric inverses weren't all that interesting, except
that the derivatives were more basic functions. So, these shouldn't be interesting
(in calculus) either, unless they provide new formulas. But they don't (at least,
none that can't be easily found other ways). However, there is an odd collection
of results: these inverses can be written easily in terms of functions we already
know, such as: 
\[
\sinh ^{-1}(x)\, =\ln (x+\sqrt{x^{2}+1}),\]
 which follows from the definitions. 

\end{document}
