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[[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|
Hyperbolic functions
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>
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[[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
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.
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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>
== History ==
The first known calculation of a hyperbolic trigonometry problem is attributed to [[Gerardus Mercator]] when issuing the [[Mercator projection|Mercator map projection]] circa 1566. It requires tabulating solutions to a [[transcendental equation]] involving hyperbolic functions.<ref name=":3">{{Cite book |last=George F. Becker |url=https://archive.org/details/hyperbolicfuncti027682mbp/page/n49/mode/2up?q=mercator |title=Hyperbolic Functions |last2=C. E. Van Orstrand |date=1909 |publisher=The Smithsonian Institution |others=Universal Digital Library}}</ref>
The first to suggest a similarity between the sector of the circle and that of the hyperbola was [[Isaac Newton]] in his 1687 [[Philosophiæ Naturalis Principia Mathematica|''Principia Mathematica'']].<ref name=":0">{{Cite book |last=McMahon |first=James |url=https://archive.org/details/hyperbolicfuncti031883mbp/page/n75/mode/2up |title=Hyperbolic Functions |date=1896 |publisher=John Wiley And Sons |others=Osmania University, Digital Library Of India}}</ref>
[[Roger Cotes]] suggested to modify the trigonometric functions using the [[imaginary unit]] <math>i=\sqrt{-1} </math> to obtain an oblate [[spheroid]] from a prolate one.<ref name=":0" />
Hyperbolic functions were formally introduced in 1757 by [[Vincenzo Riccati]].<ref name=":0" /><ref name=":3" /><ref name=":4" /> Riccati used {{math|''Sc.''}} and {{math|''Cc.''}} ({{lang|la|sinus/cosinus circulare}}) to refer to circular functions and {{math|''Sh.''}} and {{math|''Ch.''}} ({{lang|la|sinus/cosinus hyperbolico}}) to refer to hyperbolic functions.<ref name=":0" /> As early as 1759, [[François Daviet de Foncenex|Daviet de Foncenex]] showed the interchangeability of the trigonometric and hyperbolic functions using the imaginary unit and extended [[de Moivre's formula]] to hyperbolic functions.<ref name=":4" /><ref name=":0" />
During the 1760s, [[Johann Heinrich Lambert]] systematized the use functions and provided exponential expressions in various publications.<ref name=":0" /><ref name=":4">Bradley, Robert E.; D'Antonio, Lawrence A.; Sandifer, Charles Edward. ''Euler at 300: an appreciation.'' Mathematical Association of America, 2007. Page 100.</ref> Lambert credited Riccati for the terminology and names of the functions, but altered the abbreviations to those used today.<ref name=":4" /><ref>Becker, Georg F. ''Hyperbolic functions.'' Read Books, 1931. Page xlviii.</ref>
== Notation ==
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==Definitions==
[[File:Cartesian_hyperbolic_rhombus.svg|thumb|right|250px|Right triangles with legs proportional to sinh and cosh]]
[[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>]]▼
With [[hyperbolic angle]] ''u'', the hyperbolic functions sinh and cosh can defined with the [[exponential function]] e<sup>u</sup>.<ref name=":1" /><ref name=":2" /> In the figure
[[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>]]▼
<math>A =(e^{-u}, e^u), \ B=(e^u, \ e^{-u}), \ OA + OB = OC </math> .
=== 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>}}]]
* 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>
▲[[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>]]
* 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>
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===Hyperbolic tangent{{anchor|tanh}}===
The hyperbolic tangent is the (unique) solution to the [[differential equation]] {{math|1=''f'' ′ = 1 − ''f'' <sup>2</sup>}}, with {{math|1=''f'' (0) = 0}}.<ref>{{cite book |title=Nonlinear Workbook, The: Chaos, Fractals, Cellular Automata, Neural Networks, Genetic Algorithms, Gene Expression Programming, Support Vector Machine, Wavelets, Hidden Markov Models, Fuzzy Logic With C++, Java And Symbolicc++ Programs |
==Useful relations==
{{Anchor|Osborn}}
The hyperbolic functions satisfy many identities, all of them similar in form to the [[trigonometric identity|trigonometric identities]]. In fact, '''Osborn's rule'''<ref name="Osborn, 1902" /> states that one can convert any trigonometric identity (up to but not including sinhs or implied sinhs of 4th degree) for <math>\theta</math>, <math>2\theta</math>, <math>3\theta</math> or <math>\theta</math> and <math>\varphi</math> into a hyperbolic identity, by:
# expanding it completely in terms of integral powers of sines and cosines, # changing sine to sinh and cosine to cosh, and # switching the sign of every term containing a product of two sinhs. Odd and even functions:
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<math display="block">\begin{align}
\cosh x + \sinh x &= e^x \\
\cosh x - \sinh x &= e^{-x}
\cosh^2 x - \sinh^2 x &= 1▼
\end{align}</math>
which are analogous to [[Euler's formula]], and
the last of which is similar to the [[Pythagorean trigonometric identity]].▼
<math display="block">
</math>
One also has
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\end{align}</math>
Also:<ref>{{cite book|last1=Martin|first1=George E.|title=The foundations of geometry and the non-
<math display="block">\begin{align}
\sinh x - \sinh y &= 2 \cosh \left(\frac{x+y}{2}\right) \sinh \left(\frac{x-y}{2}\right)\\
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===Inequalities===
The following inequality is useful in statistics:<ref>{{cite news |last1=Audibert |first1=Jean-Yves |date=2009 |title=Fast learning rates in statistical inference through aggregation |publisher=The Annals of Statistics |page=1627}} [https://projecteuclid.org/download/pdfview_1/euclid.aos/1245332827]</ref>
It can be proved by comparing the Taylor series of the two functions term by term.
▲<math display="block">\operatorname{cosh}(x) < \frac{e^x + 1}{2}, \quad x > 0.</math>
==Inverse functions as logarithms==
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Since the [[Circular sector#Area|area of a circular sector]] with radius {{mvar|r}} and angle {{mvar|u}} (in radians) is {{math|1=''r''<sup>2</sup>''u''/2}}, it will be equal to {{mvar|u}} when {{math|1=''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 regions together depict a [[hyperbolic sector]] with area corresponding to hyperbolic angle magnitude.
The legs of the two [[right triangle]]s with [[hypotenuse]] on the ray defining the angles are of length {{radic|2}} times the circular and hyperbolic functions.
The hyperbolic angle is an [[invariant measure]] with respect to the [[squeeze mapping]], just as the circular angle is invariant under rotation.<ref>[[Mellen W. Haskell|Haskell, Mellen W.]], "On the introduction of the notion of hyperbolic functions", [[Bulletin of the American Mathematical Society]] '''1''':6:155–9, [https://www.ams.org/journals/bull/1895-01-06/S0002-9904-1895-00266-9/S0002-9904-1895-00266-9.pdf full text]</ref>
The [[Gudermannian function]] gives a direct relationship between the circular functions and the hyperbolic functions that does not involve complex numbers.
The graph of the function {{
==Relationship to the exponential function==
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\cosh(ix) &= \frac{1}{2} \left(e^{i x} + e^{-i x}\right) = \cos x \\
\sinh(ix) &= \frac{1}{2} \left(e^{i x} - e^{-i x}\right) = i \sin x \\
\tanh(ix) &= i \tan x \\
\cosh(x+iy) &= \cosh(x) \cos(y) + i \sinh(x) \sin(y) \\
\sinh(x+iy) &= \sinh(x) \cos(y) + i \cosh(x) \sin(y) \\
\cosh x &= \cos(ix) \\
\sinh x &= - i \sin(ix) \\
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