View source for Hyperbolic functions

{{complex|date=April 2020}}
{{Anchor|Sinh|Cosh|Tanh|Sech|Csch|Coth}}
[[File:sinh cosh tanh.svg|300px|thumb]]

In [[mathematics]], '''hyperbolic functions''' are analogues of the ordinary [[trigonometric function]]s, but defined for the unit [[hyperbola]] rather than on the unit [[circle]]: just as the points {{math|(cos ''t'', sin ''t'')}} form a [[unit circle|circle with a unit radius]], the points {{math|(cosh ''t'', sinh ''t'')}} form the right half of the [[hyperbola]].

Hyperbolic functions occur in the calculations of angles and distances in [[hyperbolic geometry]]. They also occur in the solutions of many linear [[differential equation]]s, [[Cubic function|cubic equations]], and [[Laplace's equation]] in [[Cartesian coordinates]]. [[Laplace's equation]]s are important in many areas of [[physics]].

The basic hyperbolic functions are:<ref name=":0">{{Cite web|date=2020-03-25|title=Comprehensive List of Algebra Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/algebra-symbols/|access-date=2020-08-29|website=Math Vault|language=en-US}}</ref><ref name=":1">{{Cite web|last=Weisstein|first=Eric W.|title=Hyperbolic Functions|url=https://mathworld.wolfram.com/HyperbolicFunctions.html|access-date=2020-08-29|website=mathworld.wolfram.com|language=en}}</ref>
* '''hyperbolic sine''' "sinh" ({{IPAc-en|ˈ|s|ɪ|ŋ|,_|ˈ|s|ɪ|n|tʃ|,_|ˈ|ʃ|aɪ|n}}),<ref>(1999) ''Collins Concise Dictionary'', 4th edition, HarperCollins, Glasgow, {{ISBN|0 00 472257 4}}, p. 1386</ref>
* '''hyperbolic cosine''' "cosh" ({{IPAc-en|ˈ|k|ɒ|ʃ|,_|ˈ|k|oʊ|ʃ}}),<ref name="Collins Concise Dictionary p. 328">''Collins Concise Dictionary'', p. 328</ref>
from which are derived:<ref name=":2">{{Cite web|title=Hyperbolic Functions|url=https://www.mathsisfun.com/sets/function-hyperbolic.html|access-date=2020-08-29|website=www.mathsisfun.com}}</ref>
* '''hyperbolic tangent''' "tanh" ({{IPAc-en|ˈ|t|æ|ŋ|,_|ˈ|t|æ|n|tʃ|,_|ˈ|θ|æ|n}}),<ref>''Collins Concise Dictionary'', p. 1520</ref>
* '''hyperbolic cosecant''' "csch" or "cosech" ({{IPAc-en|ˈ|k|oʊ|s|ɛ|tʃ|,_|ˈ|k|oʊ|ʃ|ɛ|k}}<ref name="Collins Concise Dictionary p. 328"/>)
* '''hyperbolic secant''' "sech" ({{IPAc-en|ˈ|s|ɛ|tʃ|,_|ˈ|ʃ|ɛ|k}}),<ref>''Collins Concise Dictionary'', p. 1340</ref>
* '''hyperbolic cotangent''' "coth" ({{IPAc-en|ˈ|k|ɒ|θ|,_|ˈ|k|oʊ|θ}}),<ref>''Collins Concise Dictionary'', p. 329</ref><ref>[http://www.mathcentre.ac.uk/resources/workbooks/mathcentre/hyperbolicfunctions.pdf tanh]</ref> 
corresponding to the derived trigonometric functions.

The [[inverse hyperbolic function]]s are:<ref name=":0" />
* '''area hyperbolic sine''' "arsinh" (also denoted "sinh<sup>−1</sup>", "asinh" or sometimes "arcsinh")<ref>{{Citation | last=Woodhouse | first = N. M. J. | author-link = N. M. J. Woodhouse | title = Special Relativity | publisher = Springer | place = London | date = 2003 | page = 71 | isbn = 978-1-85233-426-0}}</ref><ref>{{Citation | editor1-last=Abramowitz | editor1-first=Milton | editor1-link=Milton Abramowitz | editor2-last=Stegun | editor2-first=Irene A. | editor2-link=Irene Stegun | title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-61272-0 | year=1972| title-link=Abramowitz and Stegun }}</ref>
* '''area hyperbolic cosine''' "arcosh" (also denoted "cosh<sup>−1</sup>", "acosh" or sometimes "arccosh"
* and so on.

[[File:Hyperbolic functions-2.svg|thumb|upright=1.4|A ray through the [[unit hyperbola]] {{math|''x''<sup>2</sup> − ''y''<sup>2</sup> {{=}} 1}} in the point {{math|(cosh ''a'', sinh ''a'')}}, where {{mvar|a}} is twice the area between the ray, the hyperbola, and the {{mvar|x}}-axis. For points on the hyperbola below the {{mvar|x}}-axis, the area is considered negative (see [[:Image:HyperbolicAnimation.gif|animated version]] with comparison with the trigonometric (circular) functions).]]

The hyperbolic functions take a [[Real number|real argument]] called a [[hyperbolic angle]]. The size of a hyperbolic angle is twice the area of its [[hyperbolic sector]]. The hyperbolic functions may be defined in terms of the [[Right triangle|legs of a right triangle]] covering this sector.

Hyperbolic functions were introduced in the 1760s independently by [[Vincenzo Riccati]] and [[Johann Heinrich Lambert]].<ref>Robert E. Bradley, Lawrence A. D'Antonio, Charles Edward Sandifer. ''Euler at 300: an appreciation.'' Mathematical Association of America, 2007. Page 100.</ref>

==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>\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>\cosh x = \frac {e^x + e^{-x}} {2} = \frac {e^{2x} + 1} {2e^x} = \frac {1 + e^{-2x}} {2e^{-x}}.</math>
* Hyperbolic tangent:
*:<math>\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>\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>\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>\operatorname{csch} x = \frac{1}{\sinh x} = \frac {2} {e^x - e^{-x}}
  = \frac{2e^x} {e^{2x} - 1}</math>

==Useful relations==
The hyperbolic functions satisfy many identities, all of them similar in form to the [[trigonometric identity|trigonometric identities]]. 

Odd and even functions:

:<math>\begin{align}
  \sinh (-x) &= -\sinh x \\
  \cosh (-x) &=  \cosh x
\end{align}</math>

Hence:
:<math>\begin{align}
                \tanh (-x) &= -\tanh x \\
                \coth (-x) &= -\coth x \\
  \operatorname{sech} (-x) &=  \operatorname{sech} x \\
  \operatorname{csch} (-x) &= -\operatorname{csch} x
\end{align}</math>

Thus, {{math|cosh ''x''}} and {{math|sech ''x''}} are [[even function]]s; the others are [[odd functions]].

:<math>\begin{align}
  \operatorname{arsech} x &= \operatorname{arcosh} \left(\frac{1}{x}\right) \\
  \operatorname{arcsch} x &= \operatorname{arsinh} \left(\frac{1}{x}\right) \\
  \operatorname{arcoth} x &= \operatorname{artanh} \left(\frac{1}{x}\right)
\end{align}</math>

Hyperbolic sine and cosine satisfy: 
:<math>\begin{align}
  \cosh x + \sinh x &= e^x \\
  \cosh x - \sinh x &= e^{-x} \\
  \cosh^2 x - \sinh^2 x &= 1
\end{align}</math>

the last of which is similar to the [[Pythagorean trigonometric identity]].

One also has
:<math>\begin{align}
  \operatorname{sech} ^{2} x &= 1 - \tanh^{2} x \\
  \operatorname{csch} ^{2} x &= \coth^{2} x - 1
\end{align}</math>

for the other functions.

==Comparison with circular functions==

[[File:Circular and hyperbolic angle.svg|right|upright=1.2|thumb|Circle and hyperbola tangent at (1,1) display geometry of circular functions in terms of [[sector of a circle|circular sector]] area ''u'' and hyperbolic functions depending on [[hyperbolic sector]] area ''u''.]]
The hyperbolic functions represent an expansion of [[trigonometry]] beyond the functions defined on unit circle. Both types depend on one [[argument of a function|argument]], either [[angle|circular angle]] or [[hyperbolic angle]].

Since the [[circular sector#Area|area of a circular sector]] with radius ''r'' and angle ''u'' (in radians) is ''r''<sup>2</sup>''u''/2, it will be equal to ''u'' when ''r'' = {{sqrt|2}}. In the diagram, such a circle is tangent to the hyperbola ''xy'' = 1 at (1,1). The yellow sector depicts an area and angle magnitude. Similarly, the yellow and red sectors together depict an area and [[hyperbolic sector|hyperbolic angle magnitude]].

The legs of the two [[right triangle]]s with hypotenuse on the ray defining the angles are of length <math>\sqrt2</math> times the circular and hyperbolic functions.

==Relationship to the exponential function==

The decomposition of the exponential function in its [[even–odd decomposition|even and odd parts]] gives the identities
:<math>e^x = \cosh x + \sinh x,</math>
and
:<math>e^{-x} = \cosh x - \sinh x.</math>
The first one is analogous to [[Euler's formula]]
:<math>e^{ix} = \cos x + i\sin x.</math>

Additionally,
:<math>e^x = \sqrt{\frac{1 + \tanh x}{1 - \tanh x}} = \frac{1 + \tanh \frac{x}{2}}{1 - \tanh \frac{x}{2}}</math>

==Related pages==
* [[e (mathematical constant)]]
* [[Trigonometric functions]]

==References==
{{Reflist|35em|refs=
<!-- 
<ref name="Osborn, 1902" >{{Cite journal
  |first=G.  |last=Osborn
  |jstor=3602492 
  |title=Mnemonic for hyperbolic formulae
  |journal=[[The Mathematical Gazette]]
  |page=189
  |volume=2  |issue=34
  |date=July 1902
|doi=10.2307/3602492
 |url=https://zenodo.org/record/1449741
 }}</ref> -->

}}

==Other websites==
{{Commons category|Hyperbolic functions}}
*[http://planetmath.org/hyperbolicfunctions Hyperbolic functions] on [[PlanetMath]]
*[https://web.archive.org/web/20071006172054/http://glab.trixon.se/ GonioLab]: Visualization of the unit circle, trigonometric and hyperbolic functions ([[Java Web Start]])
*[http://www.calctool.org/CALC/math/trigonometry/hyperbolic Web-based calculator of hyperbolic functions]

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[[Category:Functions and mappings]]
[[Category:Trigonometry]]