View source for Hyperbolic functions

{{Short description|Collective name of 6 mathematical functions}}
{{Anchor|Sinh|Cosh|Tanh|Sech|Csch|Coth}}
[[File:sinh cosh tanh.svg|300px|thumb]]

In [[Mathematics|mathematics]], '''hyperbolic functions''' are analogues of the ordinary trigonometric functions, but defined using the [[Hyperbola|hyperbola]] rather than the [[Circle|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|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 occur in the calculations of angles and distances in [[Hyperbolic geometry|hyperbolic geometry]]. They also occur in the solutions of many linear [[Differential equation|differential equation]]s (such as the equation defining a [[Catenary|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:Physics|physics]], including electromagnetic theory, [[Physics:Heat transfer|heat transfer]], [[Physics:Fluid dynamics|fluid dynamics]], and [[Physics:Special relativity|special relativity]].

The basic hyperbolic functions are:<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''' "{{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:
* '''area 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. |  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=[[Social:Dover Publications|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>
* '''area hyperbolic cosine''' "{{math|arcosh}}" (also denoted "{{math|cosh<sup>−1</sup>}}", "{{math|acosh}}" or sometimes "{{math|arccosh}}")
* '''area hyperbolic tangent''' "{{math|artanh}}" (also denoted "{{math|tanh<sup>−1</sup>}}", "{{math|atanh}}" or sometimes "{{math|arctanh}}")
* '''area hyperbolic cotangent''' "{{math|arcoth}}" (also denoted "{{math|coth<sup>−1</sup>}}", "{{math|acoth}}" or sometimes "{{math|arccoth}}")
* '''area hyperbolic secant''' "{{math|arsech}}" (also denoted "{{math|sech<sup>−1</sup>}}", "{{math|asech}}" or sometimes "{{math|arcsech}}")
* '''area 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 through the [[Unit hyperbola|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 animated version with comparison with the trigonometric (circular) functions).]]

The hyperbolic functions take a [[Real number|real argument]] called a [[Hyperbolic angle|hyperbolic angle]]. The size of a hyperbolic angle is twice the area of its [[Hyperbolic sector|hyperbolic sector]]. 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|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|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>

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> 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. Lambert adopted the names, but altered the abbreviations to those used today.<ref>Georg F. Becker. ''Hyperbolic functions.'' Read Books, 1931. Page xlviii.</ref> The abbreviations {{math|sh}}, {{math|ch}}, {{math|th}}, {{math|cth}} are also currently used, depending on personal preference.

== Notation ==

==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|exponential function]]:<ref name=":1" /><ref name=":2" />

* Hyperbolic sine: the 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 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|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 functions 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|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).

== Characterizing properties==

=== Hyperbolic cosine ===

It can be shown that the area under the curve of the hyperbolic cosine (over a finite interval) is always equal to the [[Arc length|arc length]] corresponding to that interval:<ref>{{cite book | title=Golden Integral Calculus | first1=Bali | last1=N.P. | publisher=Firewall Media | year=2005 | isbn=81-7008-169-6 | page=472 | url=https://books.google.com/books?id=hfi2bn2Ly4cC&pg=PA472}}</ref>
<math display="block">\text{area} = \int_a^b \cosh x \,dx = \int_a^b \sqrt{1 + \left(\frac{d}{dx} \cosh x \right)^2} \,dx = \text{arc length.}</math>

===Hyperbolic tangent{{anchor|tanh}}===

The hyperbolic tangent is the (unique) solution to the [[Differential equation|differential equation]] {{math|1=''f''&thinsp;′ = 1 − ''f''&thinsp;<sup>2</sup>}}, with {{math|1=''f''&hairsp;(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 |author1=Willi-hans Steeb |edition= 3rd|publisher=World Scientific Publishing Company |year=2005 |isbn=978-981-310-648-2 |page=281 |url=https://books.google.com/books?id=-Qo8DQAAQBAJ}} [https://books.google.com/books?id=-Qo8DQAAQBAJ&pg=PA281 Extract of page 281 (using lambda=1)]</ref><ref>{{cite book |title=An Atlas of Functions: with Equator, the Atlas Function Calculator |author1=Keith B. Oldham |author2=Jan Myland |author3=Jerome Spanier |edition=2nd, illustrated |publisher=Springer Science & Business Media |year=2010 |isbn=978-0-387-48807-3 |page=290 |url=https://books.google.com/books?id=UrSnNeJW10YC}} [https://books.google.com/books?id=UrSnNeJW10YC&pg=PA290 Extract of page 290]</ref>

==Useful relations==
{{Anchor|Osborn}}
The hyperbolic functions satisfy many identities, all of them similar in form to the 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:
<math display="block">\begin{align}
 \sinh (-x) &= -\sinh x \\
 \cosh (-x) &=  \cosh x
\end{align}</math>

Hence:
<math display="block">\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 functions; the others are odd functions.

<math display="block">\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 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>

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

One also has
<math display="block">\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.

===Sums of arguments===
<math display="block">\begin{align}
 \sinh(x + y) &= \sinh x \cosh y + \cosh x \sinh y \\
 \cosh(x + y) &= \cosh x \cosh y + \sinh x \sinh y \\ 
 \tanh(x + y) &= \frac{\tanh x +\tanh y}{1+ \tanh x \tanh y } \\
\end{align}</math>
particularly
<math display="block">\begin{align}
\cosh (2x) &= \sinh^2{x} + \cosh^2{x} = 2\sinh^2 x + 1 = 2\cosh^2 x - 1 \\
\sinh (2x) &= 2\sinh x \cosh x \\
\tanh (2x) &= \frac{2\tanh x}{1+ \tanh^2 x } \\
\end{align}</math>

Also:
<math display="block">\begin{align}
 \sinh x + \sinh y &= 2 \sinh \left(\frac{x+y}{2}\right) \cosh \left(\frac{x-y}{2}\right)\\
 \cosh x + \cosh y &= 2 \cosh \left(\frac{x+y}{2}\right) \cosh \left(\frac{x-y}{2}\right)\\
\end{align}</math>

===Subtraction formulas===
<math display="block">\begin{align}
 \sinh(x - y) &= \sinh x \cosh y - \cosh x \sinh y \\
 \cosh(x - y) &= \cosh x \cosh y - \sinh x \sinh y \\
 \tanh(x - y) &= \frac{\tanh x -\tanh y}{1- \tanh x \tanh y } \\
\end{align}</math>

Also:<ref>{{cite book|last1=Martin|first1=George E.|title=The foundations of geometry and the non-euclidean plane|date=1986 | publisher=Springer-Verlag|location=New York|isbn=3-540-90694-0|page=416|edition=1st corr.}}</ref>
<math display="block">\begin{align}
 \sinh x - \sinh y &= 2 \cosh \left(\frac{x+y}{2}\right) \sinh \left(\frac{x-y}{2}\right)\\
 \cosh x - \cosh y &= 2 \sinh \left(\frac{x+y}{2}\right) \sinh \left(\frac{x-y}{2}\right)\\
\end{align}</math>

===Half argument formulas===
<math display="block">\begin{align}
 \sinh\left(\frac{x}{2}\right) &= \frac{\sinh x}{\sqrt{2 (\cosh x + 1)} } &&= \sgn x \, \sqrt \frac{\cosh x - 1}{2} \\[6px]
 \cosh\left(\frac{x}{2}\right) &= \sqrt \frac{\cosh x + 1}{2}\\[6px]
 \tanh\left(\frac{x}{2}\right) &= \frac{\sinh x}{\cosh x + 1} &&= \sgn x \, \sqrt \frac{\cosh x-1}{\cosh x+1} = \frac{e^x - 1}{e^x + 1}
\end{align}</math>

where {{math|sgn}} is the [[Sign function|sign function]].

If {{math|''x'' ≠ 0}}, then<ref>{{cite web|title=Prove the identity tanh(x/2) = (cosh(x) - 1)/sinh(x) | url=https://math.stackexchange.com/q/1565753 |website=StackExchange (mathematics) | access-date=24 January 2016}}</ref>

<math display="block"> \tanh\left(\frac{x}{2}\right) = \frac{\cosh x - 1}{\sinh x} = \coth x - \operatorname{csch} x </math>
===Square formulas===
<math display="block">\begin{align}
\sinh^2 x &= \tfrac{1}{2}(\cosh 2x - 1) \\
\cosh^2 x &= \tfrac{1}{2}(\cosh 2x + 1)
\end{align}</math>

===Inequalities===

The following inequality is useful in statistics:
<math>\operatorname{cosh}(t) \leq e^{t^2 /2}</math> <ref>{{cite news |last1=Audibert |first1=Jean-Yves |date=2009 |title=Fast learning rates in statistical inference through aggregation |page=1627 |publisher=The Annals of Statistics}} [https://projecteuclid.org/download/pdfview_1/euclid.aos/1245332827]</ref>

It can be proved by comparing term by term the Taylor series of the two functions.

==Inverse functions as logarithms==

<math display="block">\begin{align}
 \operatorname {arsinh} (x) &= \ln \left(x + \sqrt{x^{2} + 1} \right) \\
 \operatorname {arcosh} (x) &= \ln \left(x + \sqrt{x^{2} - 1} \right) && x \geq 1 \\
 \operatorname {artanh} (x) &= \frac{1}{2}\ln \left( \frac{1 + x}{1 - x} \right) && | x | < 1 \\
 \operatorname {arcoth} (x) &= \frac{1}{2}\ln \left( \frac{x + 1}{x - 1} \right) && |x| > 1 \\
 \operatorname {arsech} (x) &= \ln \left( \frac{1}{x} + \sqrt{\frac{1}{x^2} - 1}\right) = \ln \left( \frac{1+ \sqrt{1 - x^2}}{x} \right) && 0 < x \leq 1 \\
 \operatorname {arcsch} (x) &= \ln \left( \frac{1}{x} + \sqrt{\frac{1}{x^2} +1}\right) && x \ne 0
\end{align}</math>

==Derivatives==
<math display="block">\begin{align}
 \frac{d}{dx}\sinh x &= \cosh x \\
 \frac{d}{dx}\cosh x &= \sinh x \\
 \frac{d}{dx}\tanh x &= 1 - \tanh^2 x = \operatorname{sech}^2 x = \frac{1}{\cosh^2 x} \\
 \frac{d}{dx}\coth x &= 1 - \coth^2 x = -\operatorname{csch}^2 x = -\frac{1}{\sinh^2 x} && x \neq 0 \\
 \frac{d}{dx}\operatorname{sech} x &= - \tanh x \operatorname{sech} x \\
 \frac{d}{dx}\operatorname{csch} x &= - \coth x \operatorname{csch} x && x \neq 0
\end{align}</math>
<math display="block">\begin{align}
 \frac{d}{dx}\operatorname{arsinh} x &= \frac{1}{\sqrt{x^2+1}} \\
 \frac{d}{dx}\operatorname{arcosh} x &= \frac{1}{\sqrt{x^2 - 1}} && 1 < x \\
 \frac{d}{dx}\operatorname{artanh} x &= \frac{1}{1-x^2} && |x| < 1 \\
 \frac{d}{dx}\operatorname{arcoth} x &= \frac{1}{1-x^2} && 1 < |x| \\
 \frac{d}{dx}\operatorname{arsech} x &= -\frac{1}{x\sqrt{1-x^2}} && 0 < x < 1 \\
 \frac{d}{dx}\operatorname{arcsch} x &= -\frac{1}{|x|\sqrt{1+x^2}} && x \neq 0
 \end{align}</math>
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==Second derivatives==
Each of the functions {{math|sinh}} and {{math|cosh}} is equal to its [[Second derivative|second derivative]], that is:
<math display="block"> \frac{d^2}{dx^2}\sinh x = \sinh x </math>
<math display="block"> \frac{d^2}{dx^2}\cosh x = \cosh x \, .</math>

All functions with this property are [[Linear combination|linear combination]]s of {{math|sinh}} and {{math|cosh}}, in particular the [[Exponential function|exponential function]]s <math> e^x </math> and <math> e^{-x} </math>.<ref>{{dlmf|id=4.34}}</ref>

==Standard integrals==

<math display="block">\begin{align}
 \int \sinh (ax)\,dx &= a^{-1} \cosh (ax) + C \\
 \int \cosh (ax)\,dx &= a^{-1} \sinh (ax) + C \\
 \int \tanh (ax)\,dx &= a^{-1} \ln (\cosh (ax)) + C \\
 \int \coth (ax)\,dx &= a^{-1} \ln \left|\sinh (ax)\right| + C \\
 \int \operatorname{sech} (ax)\,dx &= a^{-1} \arctan (\sinh (ax)) + C \\
 \int \operatorname{csch} (ax)\,dx &= a^{-1} \ln \left| \tanh \left( \frac{ax}{2} \right) \right| + C = a^{-1} \ln\left|\coth \left(ax\right) - \operatorname{csch} \left(ax\right)\right| + C = -a^{-1}\operatorname{arcoth} \left(\cosh\left(ax\right)\right) +C
\end{align}</math>

The following integrals can be proved using hyperbolic substitution:
<math display="block">\begin{align}
 \int {\frac{1}{\sqrt{a^2 + u^2}}\,du} & = \operatorname{arsinh} \left( \frac{u}{a} \right) + C \\
 \int {\frac{1}{\sqrt{u^2 - a^2}}\,du} &= \sgn{u} \operatorname{arcosh} \left| \frac{u}{a} \right| + C \\
 \int {\frac{1}{a^2 - u^2}}\,du & = a^{-1}\operatorname{artanh} \left( \frac{u}{a} \right) + C && u^2 < a^2 \\
 \int {\frac{1}{a^2 - u^2}}\,du & = a^{-1}\operatorname{arcoth} \left( \frac{u}{a} \right) + C && u^2 > a^2 \\
 \int {\frac{1}{u\sqrt{a^2 - u^2}}\,du} & = -a^{-1}\operatorname{arsech}\left| \frac{u}{a} \right| + C \\
 \int {\frac{1}{u\sqrt{a^2 + u^2}}\,du} & = -a^{-1}\operatorname{arcsch}\left| \frac{u}{a} \right| + C
\end{align}</math>

where ''C'' is the [[Constant of integration|constant of integration]].

==Taylor series expressions==
It is possible to express explicitly the [[Taylor series]] at zero (or the [[Laurent series]], if the function is not defined at zero) of the above functions.

<math display="block">\sinh x = x + \frac {x^3} {3!} + \frac {x^5} {5!} + \frac {x^7} {7!} + \cdots = \sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)!}</math>
This series is [[Convergent series|convergent]] for every [[Complex number|complex]] value of {{mvar|x}}. Since the function {{math|sinh ''x''}} is odd, only odd exponents for {{math|''x''}} occur in its Taylor series.

<math display="block">\cosh x = 1 + \frac {x^2} {2!} + \frac {x^4} {4!} + \frac {x^6} {6!} + \cdots = \sum_{n=0}^\infty \frac{x^{2n}}{(2n)!}</math>
This series is [[Convergent series|convergent]] for every [[Complex number|complex]] value of {{mvar|x}}. Since the function {{math|cosh ''x''}} is even, only even exponents for {{mvar|x}} occur in its Taylor series.

The sum of the sinh and cosh series is the infinite series expression of the [[Exponential function|exponential function]].

The following series are followed by a description of a subset of their domain of convergence, where the series is convergent and its sum equals the function.
<math display="block">\begin{align}

               \tanh x &= x - \frac {x^3} {3} + \frac {2x^5} {15} - \frac {17x^7} {315} + \cdots = \sum_{n=1}^\infty \frac{2^{2n}(2^{2n}-1)B_{2n} x^{2n-1}}{(2n)!}, \qquad \left |x \right | < \frac {\pi} {2} \\

               \coth x &= x^{-1} + \frac {x} {3} - \frac {x^3} {45} + \frac {2x^5} {945} + \cdots = \sum_{n=0}^\infty \frac{2^{2n} B_{2n} x^{2n-1}} {(2n)!}, \qquad 0 < \left |x \right | < \pi \\

 \operatorname{sech} x &= 1 - \frac {x^2} {2} + \frac {5x^4} {24} - \frac {61x^6} {720} + \cdots = \sum_{n=0}^\infty \frac{E_{2 n} x^{2n}}{(2n)!} , \qquad \left |x \right | < \frac {\pi} {2} \\

 \operatorname{csch} x &= x^{-1} - \frac {x} {6} +\frac {7x^3} {360} -\frac {31x^5} {15120} + \cdots = \sum_{n=0}^\infty \frac{ 2 (1-2^{2n-1}) B_{2n} x^{2n-1}}{(2n)!} , \qquad 0 < \left |x \right | < \pi

\end{align}</math>

where:
*<math>B_n </math> is the ''n''th [[Bernoulli number]]
*<math>E_n </math> is the ''n''th [[Euler number]]

==Infinite products and continued fractions==
The following expansions are valid in the whole complex plane:
:<math>\sinh x = x\prod_{n=1}^\infty\left(1+\frac{x^2}{n^2\pi^2}\right) =
\cfrac{x}{1 - \cfrac{x^2}{2\cdot3+x^2 -
\cfrac{2\cdot3 x^2}{4\cdot5+x^2 -
\cfrac{4\cdot5 x^2}{6\cdot7+x^2 - \ddots}}}}
</math>

:<math>\cosh x = \prod_{n=1}^\infty\left(1+\frac{x^2}{(n-1/2)^2\pi^2}\right) = \cfrac{1}{1 - \cfrac{x^2}{1 \cdot 2 + x^2 - \cfrac{1 \cdot 2x^2}{3 \cdot 4 + x^2 - \cfrac{3 \cdot 4x^2}{5 \cdot 6 + x^2 - \ddots}}}}</math>

:<math>\tanh x = \cfrac{1}{\cfrac{1}{x} + \cfrac{1}{\cfrac{3}{x} + \cfrac{1}{\cfrac{5}{x} + \cfrac{1}{\cfrac{7}{x} + \ddots}}}}</math>

==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 circular sector area {{mvar|u}} and hyperbolic functions depending on [[Hyperbolic sector|hyperbolic sector]] area {{mvar|u}}.]]
The hyperbolic functions represent an expansion of [[Trigonometry|trigonometry]] beyond the circular functions. Both types depend on an [[Argument of a function|argument]], either [[Angle|circular angle]] or [[Hyperbolic angle|hyperbolic angle]].

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|hyperbolic sector]] with area corresponding to hyperbolic angle magnitude.

The legs of the two [[Right triangle|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|invariant measure]] with respect to the [[Squeeze mapping|squeeze mapping]], just as the circular angle is invariant under rotation.<ref>Mellen W. Haskell, "On the introduction of the notion of hyperbolic functions",  [[Organization:Bulletin of the American Mathematical Society|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 {{math|''a'' cosh(''x''/''a'')}} is the [[Catenary|catenary]], the curve formed by a uniform flexible chain, hanging freely between two fixed points under uniform gravity.

==Relationship to the exponential function==

The decomposition of the exponential function in its even and odd parts gives the identities
<math display="block">e^x = \cosh x + \sinh x,</math>
and
<math display="block">e^{-x} = \cosh x - \sinh x.</math>
Combined with [[Euler's formula]]
<math display="block">e^{ix} = \cos x + i\sin x,</math>
this gives
<math display="block">e^{x+iy}=(\cosh x+\sinh x)(\cos y+i\sin y)</math>
for the general complex exponential function.

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

==Hyperbolic functions for complex numbers==
{| style="text-align:center"
|+ Hyperbolic functions in the complex plane
|[[Image:Complex Sinh.jpg|1000x140px|none]]
|[[Image:Complex Cosh.jpg|1000x140px|none]]
|[[Image:Complex Tanh.jpg|1000x140px|none]]
|[[Image:Complex Coth.jpg|1000x140px|none]]
|[[Image:Complex Sech.jpg|1000x140px|none]]
|[[Image:Complex Csch.jpg|1000x140px|none]]
|-
|<math>\sinh(z)</math>
|<math>\cosh(z)</math>
|<math>\tanh(z)</math>
|<math>\coth(z)</math>
|<math>\operatorname{sech}(z)</math>
|<math>\operatorname{csch}(z)</math>
|}
Since the [[Exponential function|exponential function]] can be defined for any [[Complex number|complex]] argument, we can also extend the definitions of the hyperbolic functions to complex arguments. The functions {{math|sinh ''z''}} and {{math|cosh ''z''}} are then [[Holomorphic function|holomorphic]].

Relationships to ordinary trigonometric functions are given by [[Euler's formula]] for complex numbers:
<math display="block">\begin{align}
  e^{i x} &= \cos x + i \sin x \\
 e^{-i x} &= \cos x - i \sin x
\end{align}</math>
so:
<math display="block">\begin{align}
   \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 \\
 \cosh(x+iy) &= \cosh(x) \cos(y) + i \sinh(x) \sin(y) \\
 \sinh(x+iy) &= \sinh(x) \cos(y) + i \cosh(x) \sin(y) \\
   \tanh(ix) &= i \tan x \\
     \cosh x &= \cos(ix) \\
     \sinh x &= - i \sin(ix) \\
     \tanh x &= - i \tan(ix)
\end{align}</math>

Thus, hyperbolic functions are [[Periodic function|periodic]] with respect to the imaginary component, with period <math>2 \pi i</math> (<math>\pi i</math> for hyperbolic tangent and cotangent).

==See also==
* [[E (mathematical constant)|e (mathematical constant)]]
* [[Equal incircles theorem]], based on sinh
* [[Hyperbolastic functions]]
* [[Hyperbolic growth]]
* Inverse hyperbolic functions
* [[List of integrals of hyperbolic functions]]
* [[Poinsot's spirals]]
* [[Sigmoid function]]
* [[Soboleva modified hyperbolic tangent]]
* [[Trigonometric functions]]

==References==
{{Reflist|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 | s2cid=125866575 | url=https://zenodo.org/record/1449741 }}</ref>

}}

==External links==
*{{springer|title=Hyperbolic functions|id=p/h048250}}
*[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]



{{DEFAULTSORT:Hyperbolic Function}}
[[Category:Exponentials]]
[[Category:Hyperbolic geometry]]
[[Category:Analytic functions]]

{{Sourceattribution|Hyperbolic functions}}