Editing Hyperbolic functions (section)

==Definitions==
[[File:sinh cosh tanh.svg|thumb|<span style="color:#b30000;">sinh</span>, <span style="color:#00b300;">cosh</span> and <span style="color:#0000b3;">tanh</span>]]
[[File:csch sech coth.svg|thumb|<span style="color:#b30000;">csch</span>, <span style="color:#00b300;">sech</span> and <span style="color:#0000b3;">coth</span>]]

There are various equivalent ways to define the hyperbolic functions.

=== Exponential definitions ===
[[File:Hyperbolic and exponential; sinh.svg|thumb|right|{{math|sinh ''x''}} is half the [[Subtraction|difference]] of {{math|''e<sup>x</sup>''}} and {{math|''e''<sup>−''x''</sup>}}]]
[[File:Hyperbolic and exponential; cosh.svg|thumb|right|{{math|cosh ''x''}} is the [[Arithmetic mean|average]] of {{math|''e<sup>x</sup>''}} and {{math|''e''<sup>−''x''</sup>}}]]

In terms of the [[exponential function]]:<ref name=":1" /><ref name=":2" />

* Hyperbolic sine: the [[odd part of a function|odd part]] of the exponential function, that is, <math display="block"> \sinh x = \frac {e^x - e^{-x}} {2} = \frac {e^{2x} - 1} {2e^x} = \frac {1 - e^{-2x}} {2e^{-x}}.</math>
* Hyperbolic cosine: the [[even part of a function|even part]] of the exponential function, that is, <math display="block"> \cosh x = \frac {e^x + e^{-x}} {2} = \frac {e^{2x} + 1} {2e^x} = \frac {1 + e^{-2x}} {2e^{-x}}.</math>
* Hyperbolic tangent: <math display="block">\tanh x = \frac{\sinh x}{\cosh x} = \frac {e^x - e^{-x}} {e^x + e^{-x}}
= \frac{e^{2x} - 1} {e^{2x} + 1}.</math>
* Hyperbolic cotangent: for {{math|''x'' ≠ 0}}, <math display="block">\coth x = \frac{\cosh x}{\sinh x} = \frac {e^x + e^{-x}} {e^x - e^{-x}}
= \frac{e^{2x} + 1} {e^{2x} - 1}.</math>
* Hyperbolic secant: <math display="block"> \operatorname{sech} x = \frac{1}{\cosh x} = \frac {2} {e^x + e^{-x}}
= \frac{2e^x} {e^{2x} + 1}.</math>
* Hyperbolic cosecant: for {{math|''x'' ≠ 0}}, <math display="block"> \operatorname{csch} x = \frac{1}{\sinh x} = \frac {2} {e^x - e^{-x}}
= \frac{2e^x} {e^{2x} - 1}.</math>

=== Differential equation definitions ===

The hyperbolic functions may be defined as solutions of [[differential equation]]s: The hyperbolic sine and cosine are the solution {{math|(''s'', ''c'')}} of the system
<math display="block">\begin{align}
c'(x)&=s(x),\\
s'(x)&=c(x),\\
\end{align}
</math>
with the initial conditions <math>s(0) = 0, c(0) = 1.</math> The initial conditions make the solution unique; without them any pair of functions <math>(a e^x + b e^{-x}, a e^x - b e^{-x})</math> would be a solution.

{{math|sinh(''x'')}} and {{math|cosh(''x'')}} are also the unique solution of the equation {{math|1=''f''&thinsp;″(''x'') = ''f''&thinsp;(''x'')}},
such that {{math|1=''f''&thinsp;(0) = 1}}, {{math|1=''f''&thinsp;′(0) = 0}} for the hyperbolic cosine, and {{math|1=''f''&thinsp;(0) = 0}}, {{math|1=''f''&thinsp;′(0) = 1}} for the hyperbolic sine.

=== Complex trigonometric definitions ===

Hyperbolic functions may also be deduced from [[trigonometric function]]s with [[complex number|complex]] arguments:

* Hyperbolic sine:<ref name=":1" /> <math display="block">\sinh x = -i \sin (i x).</math>
* Hyperbolic cosine:<ref name=":1" /> <math display="block">\cosh x = \cos (i x).</math>
* Hyperbolic tangent: <math display="block">\tanh x = -i \tan (i x).</math>
* Hyperbolic cotangent: <math display="block">\coth x = i \cot (i x).</math>
* Hyperbolic secant: <math display="block"> \operatorname{sech} x = \sec (i x).</math>
* Hyperbolic cosecant:<math display="block">\operatorname{csch} x = i \csc (i x).</math>
where {{mvar|i}} is the [[imaginary unit]] with {{math|1=''i''<sup>2</sup> = −1}}.

The above definitions are related to the exponential definitions via [[Euler's formula]] (See {{Section link||Hyperbolic functions for complex numbers}} below).