Editing Hyperbolic functions (section)

{{Short description|Collective name of 6 mathematical functions}}
{{Redirect|Hyperbolic curve|the geometric curve|Hyperbola}}
{{Anchor|Sinh|Cosh|Tanh|Sech|Csch|Coth}}
[[File:sinh cosh tanh.svg|333x333px|thumb]]

In [[mathematics]], '''hyperbolic functions''' are analogues of the ordinary [[trigonometric function]]s, but defined using the [[hyperbola]] rather than the [[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 [[unit hyperbola]]. Also, similarly to how the derivatives of {{math|sin(''t'')}} and {{math|cos(''t'')}} are {{math|cos(''t'')}} and {{math|–sin(''t'')}} respectively, the derivatives of {{math|sinh(''t'')}} and {{math|cosh(''t'')}} are {{math|cosh(''t'')}} and {{math|sinh(''t'')}} respectively.

Hyperbolic functions are used to express the [[angle of parallelism]] in [[hyperbolic geometry]]. They are used to express [[Lorentz boost]]s as [[hyperbolic rotation]]s in [[special relativity]]. They also occur in the solutions of many linear [[differential equation]]s (such as the equation defining a [[catenary]]), [[Cubic equation#Hyperbolic solution for one real root|cubic equations]], and [[Laplace's equation]] in [[Cartesian coordinates]]. [[Laplace's equation]]s are important in many areas of [[physics]], including [[electromagnetic theory]], [[heat transfer]], and [[fluid dynamics]].

The basic hyperbolic functions are:<ref name=":1">{{Cite web|last=Weisstein|first=Eric W.|authorlink=Eric W. Weisstein|title=Hyperbolic Functions| url=https://mathworld.wolfram.com/HyperbolicFunctions.html|access-date=2020-08-29|website=mathworld.wolfram.com|language=en}}</ref>
* '''hyperbolic sine''' "{{math|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''' "{{math|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''' "{{math|tanh}}" ({{IPAc-en|ˈ|t|æ|ŋ|,_|ˈ|t|æ|n|tʃ|,_|ˈ|θ|æ|n}}),<ref>''Collins Concise Dictionary'', p. 1520</ref>
* '''hyperbolic cotangent''' "{{math|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>
* '''hyperbolic secant''' "{{math|sech}}" ({{IPAc-en|ˈ|s|ɛ|tʃ|,_|ˈ|ʃ|ɛ|k}}),<ref>''Collins Concise Dictionary'', p. 1340</ref>
* '''hyperbolic cosecant''' "{{math|csch}}" or "{{math|cosech}}" ({{IPAc-en|ˈ|k|oʊ|s|ɛ|tʃ|,_|ˈ|k|oʊ|ʃ|ɛ|k}}<ref name="Collins Concise Dictionary p. 328"/>)
corresponding to the derived trigonometric functions.

The [[inverse hyperbolic functions]] are:
* '''inverse hyperbolic sine''' "{{math|arsinh}}" (also denoted "{{math|sinh<sup>−1</sup>}}", "{{math|asinh}}" or sometimes "{{math|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><ref>[https://www.google.com/books?q=arcsinh+-library Some examples of using '''arcsinh'''] found in [[Google Books]].</ref>
* '''inverse hyperbolic cosine''' "{{math|arcosh}}" (also denoted "{{math|cosh<sup>−1</sup>}}", "{{math|acosh}}" or sometimes "{{math|arccosh}}")
* '''inverse hyperbolic tangent''' "{{math|artanh}}" (also denoted "{{math|tanh<sup>−1</sup>}}", "{{math|atanh}}" or sometimes "{{math|arctanh}}")
* '''inverse hyperbolic cotangent''' "{{math|arcoth}}" (also denoted "{{math|coth<sup>−1</sup>}}", "{{math|acoth}}" or sometimes "{{math|arccoth}}")
* '''inverse hyperbolic secant''' "{{math|arsech}}" (also denoted "{{math|sech<sup>−1</sup>}}", "{{math|asech}}" or sometimes "{{math|arcsech}}")
* '''inverse hyperbolic cosecant''' "{{math|arcsch}}" (also denoted "{{math|arcosech}}", "{{math|csch<sup>−1</sup>}}", "{{math|cosech<sup>−1</sup>}}","{{math|acsch}}", "{{math|acosech}}", or sometimes "{{math|arccsch}}" or "{{math|arccosech}}")
[[File:Hyperbolic functions-2.svg|thumb|upright=1.4|A [[Ray (geometry)|ray]] through the [[unit hyperbola]] {{math|1=''x''<sup>2</sup> − ''y''<sup>2</sup> = 1}} at 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 of a function|argument]] called a [[hyperbolic angle]]. The magnitude of a hyperbolic angle is the [[area]] of its [[hyperbolic sector]] to ''xy'' = 1. The hyperbolic functions may be defined in terms of the [[hyperbolic sector#Hyperbolic triangle|legs of a right triangle]] covering this sector.

In [[complex analysis]], the hyperbolic functions arise when applying the ordinary sine and cosine functions to an imaginary angle. The hyperbolic sine and the hyperbolic cosine are [[entire function]]s. As a result, the other hyperbolic functions are [[meromorphic function|meromorphic]] in the whole complex plane.

By [[Lindemann–Weierstrass theorem]], the hyperbolic functions have a [[transcendental number|transcendental value]] for every non-zero [[algebraic number|algebraic value]] of the argument.<ref>{{Cite book | jstor=10.4169/j.ctt5hh8zn| title=Irrational Numbers | volume=11| last1=Niven| first1=Ivan| year=1985| publisher=Mathematical Association of America| isbn=9780883850381}}</ref>