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\begin{document}

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\section*{hyperbolic function}

[[!redirects hyperbolic functions]]

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{trigonometry}{}\paragraph*{{Trigonometry}}\label{trigonometry}

[[!include trigonometry -- contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{alternative_presentation}{Alternative presentation}\dotfill \pageref*{alternative_presentation} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The hyperbolic functions (or sometimes \emph{hyperbolic trigonometric functions}) are analogues of the usual [[trigonometric functions]], but adapted to the [[hyperbola]] $x^2 - y^2 = 1$ rather than the [[circle]] $x^2 + y^2 = 1$.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

There are multiple ways of introducing the hyperbolic functions. Probably the most straightforward is to use the following definitions based on the [[exponential function]] $\exp$:

\begin{itemize}%
\item $\cosh(x) \coloneqq \frac1{2} (\exp(x) + \exp(-x))$ (\emph{hyperbolic cosine}, sometimes pronounced as ``kosh''). This can be interpreted as a function $\mathbb{R} \to \mathbb{R}$, or as a function $\mathbb{C} \to \mathbb{C}$.


\item $\sinh(x) \coloneqq \frac1{2} (\exp(x) - \exp(-x))$ (\emph{hyperbolic sine}, sometimes pronounced as ``cinch''). This also can be interpreted as a function $\mathbb{R} \to \mathbb{R}$, or as a function $\mathbb{C} \to \mathbb{C}$.


\item The remaining hyperbolic functions are defined by $\tanh \coloneqq \frac{\sinh}{\cosh}$, $\coth = \frac{\cosh}{\sinh}$, $\sech = \frac1{\cosh}$, $\csch = \frac1{\sinh}$.



\end{itemize}
Note that $(\cosh(t), \sinh(t))$ (as a pair of real-valued functions) lies on the hyperbola $x^2 - y^2 = 1$, and in fact this is a parametrization of the hyperbola, much as $(\cos(t), \sin(t))$ parametrizes the unit circle $x^2 + y^2 = 1$.

It is straightforward to establish the following further properties by exploiting properties of the exponential function:

\begin{itemize}%
\item $\cosh(x + y) = \cosh(x)\cosh(y) + \sinh(x)\sinh(y)$ and $\sinh(x+y) = \sinh(x)\cosh(y) - \cosh(x)\sinh(y)$ (``addition formulas''),


\item $(\cosh)' = \sinh$ and $(\sinh)' = \cosh$,


\item $\cosh(x + 2\pi i) = \cosh(x)$ and $\sinh(x + 2\pi i) = \sinh(x)$ for all $x \in \mathbb{C}$,


\item $\cosh(x) = \sum_{n \geq 0} \frac{x^{2 n}}{(2 n)!}$ and $\sinh(x) = \sum_{n \geq 0} \frac{x^{2 n + 1}}{(2 n + 1)!}$.



\end{itemize}
\hypertarget{alternative_presentation}{}\subsubsection*{{Alternative presentation}}\label{alternative_presentation}

Another avenue is first to introduce the \emph{inverse} hyperbolic functions as [[integrals]] of suitable [[algebraic functions]], e.g.,

\begin{displaymath}
\sinh^{-1}(t) = \int_0^t \; \frac{d x}{y}
\end{displaymath}
where $y = \sqrt{x^2 + 1}$. Or, what is essentially the same, construct a solution $p(t)$ to the [[differential equation]]

\begin{displaymath}
(p')^2 = p^2 + 1, \qquad p(0) = 0
\end{displaymath}
so that $(p'(t), p(t))$ parametrizes the curve $x^2 - y^2 = 1$; the approach by invoking an integral is in accordance with an elementary method in differential equations that goes by the name ``separation of variables''.

This particular approach is similar to the way that [[elliptic functions]] were introduced historically, by studying integrals of algebraic functions

\begin{displaymath}
\int \; \frac{d x}{y}
\end{displaymath}
where $y$ is related to $x$ via an algebraic relation such as $y^2 = x^3 + a x + b$. For suitable such relations these give the various so-called \emph{elliptic integrals}; for more on what this has to do with ellipses, see Wikipedia, e.g., \href{http://en.wikipedia.org/wiki/Elliptic_integral#Incomplete_elliptic_integral_of_the_first_kind}{here}. Elliptic functions are then suitable inverses of elliptic integrals, following Jacobi, Abel, and others throughout the nineteenth century (e.g., Weierstrass; see also [[Weierstrass elliptic function]]).

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[trigonometry]]

\end{itemize}
[[!redirects hyperbolic functions]]

[[!redirects hyperbolic trigonometric function]] [[!redirects hyperbolic trigonometric functions]]



\end{document}