Hyperbolic functions: Difference between revisions

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{{Short description|Collective name of 6 mathematical functions}}

In [[mathematics]], the '''hyperbolic functions''' are analogs of the ordinary [[trigonometric function|trigonometric]], or circular, 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.

Given <math>\imath \equiv \sqrt {-1}</math> (See [[Complex Numbers]]), these functions are:

<div style="float:right;width:201;margin-left:0.5em;">

[[image:sinh cosh tanh.png|Hyperbolic functions look like curves]]<br>

<font color=#b30000>sinh</font>, <font color=#00b300>cosh</font> and <font color=#0000b3>tanh</font><br>

[[image:csch sech coth.png|Hyperbolic functions look like curves]]<br>

<font color=#b30000>csch</font>, <font color=#00b300>sech</font> and <font color=#0000b3>coth</font>

</div>

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]].

:<math>\sinh(x) = \frac{e^x - e^{-x}}{2} = -\imath \sin(\imath x)</math>

::(''hyperbolic sine'', pronounced "shine" or "sinch")

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>

:<math>\cosh(x) = \frac{e^{x} + e^{-x}}{2} = \cos(\imath x) </math>

* '''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'', pronounced "cosh")

* '''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:

:<math>\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac {e^x - e^{-x}} {e^x + e^{-x}} = -\imath \tan(\imath x)</math>

* '''inverse hyperbolic sine''' "{{math|arsinh}}" (also denoted "{{math|sinh⁻¹}}", "{{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>

::(''hyperbolic tangent'', pronounced "than" or "tanch")

* '''inverse hyperbolic cosine''' "{{math|arcosh}}" (also denoted "{{math|cosh⁻¹}}", "{{math|acosh}}" or sometimes "{{math|arccosh}}")

* '''inverse hyperbolic tangent''' "{{math|artanh}}" (also denoted "{{math|tanh⁻¹}}", "{{math|atanh}}" or sometimes "{{math|arctanh}}")

* '''inverse hyperbolic cotangent''' "{{math|arcoth}}" (also denoted "{{math|coth⁻¹}}", "{{math|acoth}}" or sometimes "{{math|arccoth}}")

* '''inverse hyperbolic secant''' "{{math|arsech}}" (also denoted "{{math|sech⁻¹}}", "{{math|asech}}" or sometimes "{{math|arcsech}}")

* '''inverse hyperbolic cosecant''' "{{math|arcsch}}" (also denoted "{{math|arcosech}}", "{{math|csch⁻¹}}", "{{math|cosech⁻¹}}","{{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''² − ''y''² = 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.

:<math>\coth(x) = \frac{\cosh(x)}{\sinh(x)} = \frac {e^x + e^{-x}} {e^x - e^{-x}} = \imath \cot(\imath x)</math>

::(''hyperbolic cotangent'', pronounced "coth" or "chot")

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.

:<math>\operatorname{sech}(x) = \frac{1}{\cosh(x)} = \frac {2} {e^x + e^{-x}} = \sec(\imath x)</math>

::(''hyperbolic secant'', pronounced "sheck" or "sech")

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>

:<math>\operatorname{csch}(x) = \frac{1}{\sinh(x)} = \frac {2} {e^x - e^{-x}} = \imath \csc(\imath x)</math>

::(''hyperbolic cosecant'', pronounced "cosheck" or "cosech")

==Series definitionHistory ==

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>

It is possible to express the above functions as series:

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>

:<math>\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>

[[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" />

:<math>\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>

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" />

:<math>\tanh x = x - \frac {x^3} {3} + \frac {2x^5} {15} - \frac {17x^7} {315} + \cdots = \sum_{n=1}^\infty \frac{(-1)^{n-1}2^{2n}(2^{2n}-1)B_nx^{2n-1}}{(2n)!}, \left |x \right | < \frac {\pi} {2} </math>

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>

:<math>\coth x = \frac {1} {x} + \frac {x} {3} - \frac {x^3} {45} + \frac {2x^5} {945} + \cdots = \frac {1} {x} + \sum_{n=1}^\infty \frac{(-1)^{n-1}2^{2n} B_n x^{2n-1}} {(2n)!}, 0 < \left |x \right | < \pi </math>

== Notation ==

:<math>\operatorname {sech} x = 1 - \frac {x^2} {2} + \frac {5x^4} {24} - \frac {61x^6} {720} + \cdots = 1 + \sum_{n=1}^\infty \frac{(-1)^n E_n x^{2n}}{(2n)!} , \left |x \right | < \frac {\pi} {2} </math>

{{main|Trigonometric functions#Notation}}

==Definitions==

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

[[File:Cartesian_hyperbolic_rhombus.svg|thumb|right|250px|Right triangles with legs proportional to sinh and cosh]]

With [[hyperbolic angle]] ''u'', the hyperbolic functions sinh and cosh can defined with the [[exponential function]] e^u.<ref name=":1" /><ref name=":2" /> In the figure

<math>A =(e^{-u}, e^u), \ B=(e^u, \ e^{-u}), \ OA + OB = OC </math> .

=== Exponential definitions ===

where

[[File:Hyperbolic and exponential; sinh.svg|thumb|right|{{math|sinh ''x''}} is half the [[Subtraction|difference]] of {{math|''e^x''}} and {{math|''e''^−''x''}}]]

[[File:Hyperbolic and exponential; cosh.svg|thumb|right|{{math|cosh ''x''}} is the [[Arithmetic mean|average]] of {{math|''e^x''}} and {{math|''e''^−''x''}}]]

* 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>

:<math>B_n \,</math> is the nth [[Bernoulli number]]

* 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>

:<math>E_n \,</math> is the nth [[Euler number]]

[[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>

* 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 ===

==Relationship to regular trigonometric functions==

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

Just as the points (cos ''t'', sin ''t'') define a [[circle]], the points (cosh ''t'', sinh ''t'') define the right half of the equilateral [[hyperbola]] ''x''&sup2; - ''y''&sup2; = 1. This is based on the easily verified identity

<math display="block">\begin{align}

:<math>\cosh^2(t) - \sinh^2(t) = 1 \,</math>

c'(x)&=s(x),\\

and the property that cosh ''t > 0'' for all ''t''.

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'')}},

However, the hyperbolic functions are not [[periodic function|periodic]].

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 ===

The parameter ''t'' is not a circular [[angle]], but rather a [[hyperbolic angle]] which represents twice the area between the ''x''-axis, the hyperbola and the straight line which links the origin with the point (cosh ''t'', sinh ''t'') on the hyperbola.

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

The function cosh ''x'' is an [[even function]], that is symmetric with respect to the ''y''-axis, and cosh&nbsp;0&nbsp;=&nbsp;1.

* Hyperbolic sine:<ref name=":1" /> <math display="block">\sinh x = -i \sin (i x).</math>

The function sinh ''x'' is an [[odd function]], that is symmetric with respect to the origin, and sinh&nbsp;0&nbsp;=&nbsp;0.

* 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''² = −1}}.

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

The hyperbolic functions satisfy many identities, all of them similar in form to the [[trigonometric identity|trigonometric identities]]. In fact, '''Osborne's rule''' states that one can convert any trigonometric identity 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 which contains a product of two sinh's. This yields for example the addition theorems

:<math>\sinh(x+y) = \sinh(x) \cosh(y) + \cosh(x) \sinh(y) \,</math>

:<math>\cosh(x+y) = \cosh(x) \cosh(y) + \sinh(x) \sinh(y) \,</math>

and the "half-angle formulas"

:<math>\cosh^2\left(\frac{x}{2}\right) = \frac{1+\cosh(x)}{2}</math>

:<math>\sinh^2\left(\frac{x}{2}\right) = \frac{\cosh(x)-1}{2}</math>

== Characterizing properties==

The [[derivative]] of sinh ''x'' is given by cosh ''x'' and the derivative of cosh ''x'' is sinh ''x''.

=== Hyperbolic cosine ===

The graph of the function cosh ''x'' is the [[catenary]] curve.

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]] 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>

==Inverse hyperbolic functions==

<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}}===

[[Image:Area_tangens.png|thumb|200px|right|Arctanh function]]

The [[inverse function|inverse]]s of the hyperbolic functions are often called the arc hyperbolic functions:

The hyperbolic tangent is the (unique) solution to the [[differential equation]] {{math|1=''f''&thinsp;′ = 1 − ''f''&thinsp;²}}, 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 |first=Willi-Hans |last= 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 |first1=Keith B.|last1= Oldham |first2=Jan |last2=Myland |first3=Jerome |last3=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>

:<math>\operatorname{arcsinh}(x) = \ln(x + \sqrt{x^2 + 1})</math>

:<math>\operatorname{arccosh}(x) = \ln(x \pm \sqrt{x^2 - 1})</math>

:<math>\operatorname{arctanh}(x) = \ln\left(\frac{\sqrt{1 - x^2}}{1-x}\right) = \begin{matrix} \frac{1}{2} \end{matrix} \ln\left(\frac{1+x}{1-x}\right)</math>

:<math>\operatorname{arccoth}(x) = \ln\left(\frac{\sqrt{x^2 - 1}}{x-1}\right) = \begin{matrix} \frac{1}{2} \end{matrix} \ln\left(\frac{x+1}{x-1}\right)</math>

:<math>\operatorname{arcsech}(x) = \ln\left(\frac{1 \pm \sqrt{1 - x^2}}{x}\right)</math>

:<math>\operatorname{arccsch}(x) = \ln\left(\frac{1 \pm \sqrt{1 + x^2}}{x}\right)</math>

==Useful relations==

==Applications of hyperbolic functions to integrals==

{{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:

:<math>\int \frac {dx} {\sqrt{1 - x^2}} = \operatorname{arcsin}(x)+{C} = - \operatorname{arccos}(x) + \frac {\pi}{2}+{C}</math>

<math display="block">\begin{align}

\sinh (-x) &= -\sinh x \\

\cosh (-x) &= \cosh x

\end{align}</math>

Hence:

:<math>\int \frac {dx} {\sqrt{x^2 + 1}} = \operatorname{arcsinh}(x)+{C} = \ln(x + \sqrt{x^2 + 1})+{C}</math>

<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 function]]s; the others are [[odd functions]].

:<math>\int \frac {dx} {\sqrt{x^2 - 1}} = \operatorname{arccosh}(x)+{C} = \ln(x + \sqrt{x^2 - 1})+{C}</math>

<math display="block">\begin{align}

:<math>\int \sqrt{1 - x^2} dx = \frac{\operatorname{arcsin}(x) + x\sqrt{1 - x^2}}{2}+{C}</math>

\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>\int \sqrt{x^2 + 1} dx = \frac{\operatorname{arcsinh}(x) + x\sqrt{x^2 + 1}}{2}+{C} = \frac{\ln(x + \sqrt{x^2 + 1}) + x\sqrt{x^2 + 1}}{2}+{C}</math>

<math display="block">\begin{align}

\cosh x + \sinh x &= e^x \\

\cosh x - \sinh x &= e^{-x}

\end{align}</math>

which are analogous to [[Euler's formula]], and

:<math>\int \sqrt{x^2 - 1} dx = \frac{- \operatorname{arccos}h(x) + x\sqrt{x^2 - 1}}{2}+{C} = \frac{- \ln(x + \sqrt{x^2 - 1}) + x\sqrt{x^2 - 1}}{2}+{C}</math>

<math display="block">

:<math>\int \frac {dx} {1 + x^2} = \operatorname{arctan}(x)+{C}</math>

\cosh^2 x - \sinh^2 x = 1

</math>

which is analogous to the [[Pythagorean trigonometric identity]].

:<math>\int \frac {dx} {1 - x^2} = \operatorname{arctanh}(x) = \begin{matrix} \frac{1}{2} \end{matrix} \ln\left(\frac{1+x}{1-x}\right)+{C}</math>

One also has

==Hyperbolic functions for complex numbers==

<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.

Since the [[exponential function]] can be defined for any [[complex number|complex]] argument, we can extend the definitions of the hyperbolic functions also to complex arguments. The functions sinh ''z'' and cosh ''z'' are then [[holomorphic]]; their Taylor series expansions are given in the [[Taylor series]] article.

===Sums of arguments===

Relationships to regular trigonometric functions are given by [[Euler's formula]] for complex numbers:

<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>e^{\imath x} = \cos x + \imath \;\sin x</math>

<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>\cosh(\imath x) = \frac{(e^{\imath x} + e^{-\imath x})}{2} = \cos(x)</math>

<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>\sinh(\imath x) = \frac{(e^{\imath x} - e^{-\imath x})}{2} = \imath \sin(x)</math>

<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>\tanh(\imath x) = \imath \tan(x) \,</math>

<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]].

:<math>\sinh(x) = -\imath \sin(\imath x) \,</math>

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>\cosh(x) = \cos(\imath x) \,</math>

:<math display="block"> \tanh\left(\frac{x}{2}\right) = \frac{\cosh x - 1}{\imathsinh x} = \tan(\imathcoth 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===

:<math>\operatorname{arcsinh}(x) = \imath \arcsin(-\imath x)</math>

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>

:<math>\operatorname{arccosh}(x) = \imath \arccos(x)</math>

<math display="block">\operatorname{cosh}(t) \leq e^{t^2 /2}.</math>

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

:<math>\operatorname{arctanh}(x) = \imath \arctan(-\imath x)</math>

:<math>\ 2\sum_{j=n}^{kn-1} \operatorname{arctanh}\left(\frac{1}{1 + 2\,j}\right)=\ln k </math>

==Inverse functions as logarithms==

{{main|Inverse hyperbolic function}}

<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]], 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]]s of {{math|sinh}} and {{math|cosh}}, in particular the [[exponential function]]s <math> e^x </math> and <math> e^{-x} </math>.<ref>{{dlmf|id=4.34}}</ref>

==Standard integrals==

{{For|a full list|list of integrals of hyperbolic functions}}

<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]].

==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 function|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 function|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]].

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 [[sector of a circle|circular sector]] area {{mvar|u}} and hyperbolic functions depending on [[hyperbolic sector]] area {{mvar|u}}.]]

The hyperbolic functions represent an expansion of [[trigonometry]] beyond the [[circular function]]s. Both types depend on an [[argument of a function|argument]], either [[angle|circular angle]] or [[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''²''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 {{tmath|a\cosh (x/a)}} is the [[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–odd decomposition|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]] 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 \\

\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) \\

\tanh(x+iy) &= \frac{\tanh(x) + i \tan(y)}{1 + i \tanh(x) \tan(y)} \\

\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)]]

* [[Equal incircles theorem]], based on sinh

* [[Hyperbolastic functions]]

* [[Hyperbolic growth]]

* [[Inverse hyperbolic function]]s

* [[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==

{{Commons category|Hyperbolic functions}}

*{{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]

{{Trigonometric and hyperbolic functions}}

{{Authority control}}

{{DEFAULTSORT:Hyperbolic Function}}

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