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| [[Image:Hyperbolic functions-2.svg|thumb|296px|right|A ray through the origin intercepts the [[unit hyperbola]] <math>\scriptstyle x^2\ -\ y^2\ =\ 1</math> in the point <math>\scriptstyle (\cosh\,a,\,\sinh\,a)</math>, where <math>\scriptstyle a</math> is twice the area between the ray, the hyperbola, and the <math>\scriptstyle x</math>-axis. For points on the hyperbola below the <math>\scriptstyle x</math>-axis, the area is considered negative (see [[:Image:HyperbolicAnimation.gif|animated version]] with comparison with the trigonometric (circular) functions).]]
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| In [[mathematics]], '''hyperbolic functions''' are analogs of the ordinary [[trigonometric function|trigonometric]], or circular, functions. The basic hyperbolic functions are the '''hyperbolic sine''' "sinh" ({{IPAc-en|ˈ|s|ɪ|n|tʃ}} or {{IPAc-en|ˈ|ʃ|aɪ|n}}),<ref>(1999) ''Collins Concise Dictionary'', 4th edition, HarperCollins, Glasgow, ISBN 0 00 472257 4, p.1386</ref> and the '''hyperbolic cosine''' "cosh" ({{IPAc-en|ˈ|k|ɒ|ʃ}}),<ref>''Collins Concise Dictionary'', p.328</ref> from which are derived the '''hyperbolic tangent''' "tanh" ({{IPAc-en|ˈ|t|æ|n|tʃ}} or {{IPAc-en|ˈ|θ|æ|n}}),<ref>''Collins Concise Dictionary'', p.1520</ref> '''hyperbolic cosecant''' "csch" or "cosech" ({{IPAc-en|ˈ|k|oʊ|ʃ|ɛ|k}}<ref>''Collins Concise Dictionary'', p.328</ref> or {{IPAc-en|ˈ|k|oʊ|s|ɛ|tʃ}}), '''hyperbolic secant''' "sech" ({{IPAc-en|ˈ|ʃ|ɛ|k}} or {{IPAc-en|ˈ|s|ɛ|tʃ}}),<ref>''Collins Concise Dictionary'', p.1340</ref> and '''hyperbolic cotangent''' "coth" ({{IPAc-en|ˈ|k|oʊ|θ}} or {{IPAc-en|ˈ|k|ɒ|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 the '''area hyperbolic sine''' "arsinh" (also called "asinh" or sometimes "arcsinh")<ref>[http://www.google.com/books?q=arcsinh+-library Some examples of using '''arcsinh'''] found in [[Google Books]].</ref> and so on.
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| Just as the points (cos ''t'', sin ''t'') form a circle with a unit radius, the points (cosh ''t'', sinh ''t'') form the right half of the equilateral [[hyperbola]]. The hyperbolic functions take a [[Real number|real argument]] called a [[hyperbolic angle]]. The size of a hyperbolic angle is the area of its [[hyperbolic sector]]. The hyperbolic functions may be defined in terms of the [[hyperbolic sector#Hyperbolic triangle|legs of a right triangle]] covering this sector.
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| Hyperbolic functions occur in the solutions of some important linear [[differential equation]]s, for example the equation defining a [[catenary]], of some [[Cubic function#Trigonometric (and hyperbolic) method|cubic equations]], and of [[Laplace's equation]] in [[Cartesian coordinates]]. The latter is important in many areas of [[physics]], including [[electromagnetic theory]], [[heat transfer]], [[fluid dynamics]], and [[special relativity]].
| | '''MathML''' |
| | :<math forcemathmode="mathml">E=mc^2</math> |
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| In [[complex analysis]], the hyperbolic functions arise as the imaginary parts of sine and cosine. When considered defined by a complex variable, the hyperbolic functions are [[rational function]]s of [[exponential function|exponentials]], and are hence [[meromorphic function|meromorphic]].
| | <!--'''PNG''' (currently default in production) |
| | :<math forcemathmode="png">E=mc^2</math> |
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| 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 ''Sc.'' and ''Cc.'' (''[co]sinus circulare'') to refer to circular functions and ''Sh.'' and ''Ch.'' (''[co]sinus hyperbolico'') to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what they are today.<ref>Georg F. Becker. ''Hyperbolic functions.'' Read Books, 1931. Page xlviii.</ref> The abbreviations ''sh'' and ''ch'' are still used in some other languages, like French and Russian.
| | '''source''' |
| | :<math forcemathmode="source">E=mc^2</math> --> |
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| ==Standard algebraic expressions==
| | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
| [[Image:sinh cosh tanh.svg|256px|thumb|<span style="color:#b30000;">sinh</span>, <span style="color:#00b300;">cosh</span> and <span style="color:#0000b3;">tanh</span>]]
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| [[Image:csch sech coth.svg|256px|thumb|<span style="color:#b30000;">csch</span>, <span style="color:#00b300;">sech</span> and <span style="color:#0000b3;">coth</span>]]
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| {{multiple image
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| | footer = Hyperbolic functions (a) cosh and (b) sinh obtained using exponential functions <math>e^x</math> and <math>e^{-x}</math>
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| | image1 = Hyperbolic and exponential; cosh.svg
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| | caption1 = (a) cosh(''x'') is the [[Arithmetic mean|average]] of ''e<sup>x</sup>''and ''e<sup>−x</sup>''
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| | alt1 = (a) cosh(''x'') is the [[Arithmetic mean|average]] of ''e<sup>x</sup>''and ''e<sup>−x</sup>''
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| | image2 = Hyperbolic and exponential; sinh.svg
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| | caption2 = (b) sinh(''x'') is half the [[Subtraction|difference]] of ''e<sup>x</sup>'' and ''e<sup>−x</sup>''
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| | alt2 = (b) sinh(''x'') is half the [[Subtraction|difference]] of ''e<sup>x</sup>'' and ''e<sup>−x</sup>''
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| }}
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| The hyperbolic functions are:
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| * Hyperbolic sine:
| | ==Demos== |
| ::<math>\sinh x = \frac {e^x - e^{-x}} {2} = \frac {e^{2x} - 1} {2e^x} = \frac {1 - e^{-2x}} {2e^{-x}}</math>
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| * Hyperbolic cosine:
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
| ::<math>\cosh x = \frac {e^x + e^{-x}} {2} = \frac {e^{2x} + 1} {2e^x} = \frac {1 + e^{-2x}} {2e^{-x}}</math> | |
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| * Hyperbolic tangent:
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| ::<math>\tanh x = \frac{\sinh x}{\cosh x} = \frac {e^x - e^{-x}} {e^x + e^{-x}} = \frac{e^{2x} - 1} {e^{2x} + 1} = \frac{1 - e^{-2x}} {1 + e^{-2x}}</math>
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| * Hyperbolic cotangent: | | * accessibility: |
| ::<math>\coth x = \frac{\cosh x}{\sinh x} = \frac {e^x + e^{-x}} {e^x - e^{-x}} = \frac{e^{2x} + 1} {e^{2x} - 1} = \frac{1 + e^{-2x}} {1 - e^{-2x}}</math> | | ** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)] |
| | ** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]]. |
| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
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| * Hyperbolic secant:
| | ==Test pages == |
| ::<math>\operatorname{sech}\,x = \left(\cosh x\right)^{-1} = \frac {2} {e^x + e^{-x}} = \frac{2e^x} {e^{2x} + 1} = \frac{2e^{-x}} {1 + e^{-2x}}</math>
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| * Hyperbolic cosecant: | | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
| ::<math>\operatorname{csch}\,x = \left(\sinh x\right)^{-1} = \frac {2} {e^x - e^{-x}} = \frac{2e^x} {e^{2x} - 1} = \frac{2e^{-x}} {1 - e^{-2x}}</math> | | *[[Displaystyle]] |
| | *[[MathAxisAlignment]] |
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| | *[[Unique Ids]] |
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| Hyperbolic functions can be introduced via [[hyperbolic angle#Imaginary circular angle|imaginary circular angles]]:
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| | | *[[Url2Image|Url2Image (private Wikis only)]] |
| * Hyperbolic sine:
| | ==Bug reporting== |
| ::<math>\sinh x = -i \sin (i x)</math>
| | If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de . |
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| * Hyperbolic cosine:
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| ::<math>\cosh x = \cos (i x)</math>
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| * Hyperbolic tangent:
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| ::<math>\tanh x = -i \tan (i x)</math>
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| * Hyperbolic cotangent:
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| ::<math>\coth x = i \cot (i x)</math>
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| * Hyperbolic secant: | |
| ::<math>\operatorname{sech} x = \sec (i x)</math>
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| * Hyperbolic cosecant:
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| ::<math>\operatorname{csch} x = i \csc (i x)</math>
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| where ''i'' is the [[imaginary unit]] defined by ''i''<sup>2</sup> = −1.
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| The [[complex number|complex]] forms in the definitions above derive from [[Euler's formula]].
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| ==Useful relations==
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| Odd and even functions:
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| :<math>\begin{align}
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| \sinh (-x) &= -\sinh x \\
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| \cosh (-x) &= \cosh x
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| \end{align}</math>
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| Hence:
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| :<math>\begin{align}
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| \tanh (-x) &= -\tanh x \\
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| \coth (-x) &= -\coth x \\
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| \operatorname{sech} (-x) &= \operatorname{sech} x \\
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| \operatorname{csch} (-x) &= -\operatorname{csch} x
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| \end{align}</math>
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| It can be seen that cosh ''x'' and sech ''x'' are [[even function]]s; the others are [[odd functions]].
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| :<math>\begin{align}
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| \operatorname{arsech} x &= \operatorname{arcosh} \frac{1}{x} \\
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| \operatorname{arcsch} x &= \operatorname{arsinh} \frac{1}{x} \\
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| \operatorname{arcoth} x &= \operatorname{artanh} \frac{1}{x}
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| \end{align}</math>
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| Hyperbolic sine and cosine satisfy the identity
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| :<math>\cosh^2 x - \sinh^2 x = 1 </math>
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| which is similar to the [[Pythagorean trigonometric identity]]. One also has
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| :<math>\begin{align}
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| \operatorname{sech} ^{2} x &= 1 - \tanh^{2} x \\
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| \operatorname{csch} ^{2} x &= \coth^{2} x - 1
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| \end{align}</math>
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| for the other functions.
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| The hyperbolic tangent is the solution to the [[differential equation]] <math>f'=1-f^2</math> with f(0)=0 and the [[nonlinear]] [[boundary value problem]]:<ref>
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| {{cite web
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| | url = http://mathworld.wolfram.com/HyperbolicTangent.html
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| | title = Hyperbolic Tangent
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| | author = [[Eric W. Weisstein]]
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| | publisher = [[MathWorld]]
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| | date =
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| | accessdate = 2008-10-20
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| }}</ref>
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| :<math>\frac{1}{2} f'' = f^3 - f ; \quad f(0) = f'(\infty) = 0</math>
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| It can be shown that the area under the curve of cosh (''x'') over a finite interval is always equal to the arc length corresponding to that interval:<ref>
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| {{cite book
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| |title=Golden Integral Calculus
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| |first1=Bali
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| |last1=N.P.
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| |publisher=Firewall Media
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| |year=2005
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| |isbn=81-7008-169-6
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| |page=472
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| |url=http://books.google.com/books?id=hfi2bn2Ly4cC}}, [http://books.google.com/books?id=hfi2bn2Ly4cC&pg=PA472 Extract of page 472]
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| </ref>
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| :<math>\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>
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| Sums of arguments:
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| :<math>\begin{align}
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| \sinh(x + y) &= \sinh (x) \cosh (y) + \cosh (x) \sinh (y) \\
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| \cosh(x + y) &= \cosh (x) \cosh (y) + \sinh (x) \sinh (y) \\
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| \end{align}</math>
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| particularly
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| :<math>\begin{align}
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| \cosh (2x) &= \sinh^2{x} + \cosh^2{x} = 2\sinh^2 x + 1 = 2\cosh^2 x - 1\\
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| \sinh (2x) &= 2\sinh x \cosh x
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| \end{align}</math>
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| Sum and difference of cosh and sinh:
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| :<math>\begin{align}
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| \cosh x + \sinh x &= e^x \\
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| \cosh x - \sinh x &= e^{-x}
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| \end{align}</math>
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| ==Inverse functions as logarithms==
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| :<math>\begin{align}
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| \operatorname {arsinh} (x) &= \ln \left(x + \sqrt{x^{2} + 1} \right) \\
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| \operatorname {arcosh} (x) &= \ln \left(x + \sqrt{x^{2} - 1} \right); x \ge 1 \\
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| \operatorname {artanh} (x) &= \frac{1}{2}\ln \left( \frac{1 + x}{1 - x} \right); \left| x \right| < 1 \\
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| \operatorname {arcoth} (x) &= \frac{1}{2}\ln \left( \frac{x + 1}{x - 1} \right); \left| x \right| > 1 \\
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| \operatorname {arsech} (x) &= \ln \left( \frac{1}{x} + \frac{\sqrt{1 - x^{2}}}{x} \right); 0 < x \le 1 \\
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| \operatorname {arcsch} (x) &= \ln \left( \frac{1}{x} + \frac{\sqrt{1 + x^{2}}}{\left| x \right|} \right); x \ne 0
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| \end{align}</math>
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| ==Derivatives==
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| :<math> \frac{d}{dx}\sinh x = \cosh x \,</math>
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| :<math> \frac{d}{dx}\cosh x = \sinh x \,</math>
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| :<math> \frac{d}{dx}\tanh x = 1 - \tanh^2 x = \operatorname{sech}^2 x = 1/\cosh^2 x \,</math> <!-- from: http://www.pen.k12.va.us/Div/Winchester/jhhs/math/lessons/calculus/tableof.html and http://thesaurus.maths.org/mmkb/entry.html?action=entryById&id=2664 -->
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| :<math> \frac{d}{dx}\coth x = 1 - \coth^2 x = -\operatorname{csch}^2 x = -1/\sinh^2 x \,</math>
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| :<math> \frac{d}{dx}\ \operatorname{sech}\,x = - \tanh x \ \operatorname{sech}\,x \,</math>
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| :<math> \frac{d}{dx}\ \operatorname{csch}\,x = - \coth x \ \operatorname{csch}\,x \,</math>
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| :<math>\frac{d}{dx}\, \operatorname{arsinh}\,x =\frac{1}{\sqrt{x^{2}+1}}</math>
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| :<math>\frac{d}{dx}\, \operatorname{arcosh}\,x =\frac{1}{\sqrt{x^{2}-1}}</math>
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| :<math>\frac{d}{dx}\, \operatorname{artanh}\,x =\frac{1}{1-x^{2}}</math>
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| :<math>\frac{d}{dx}\, \operatorname{arcoth}\,x =\frac{1}{1-x^{2}}</math>
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| :<math>\frac{d}{dx}\, \operatorname{arsech}\,x =-\frac{1}{x\sqrt{1-x^{2}}}</math>
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| :<math>\frac{d}{dx}\, \operatorname{arcsch}\,x =-\frac{1}{\left| x \right|\sqrt{1+x^{2}}}</math>
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| ==Standard integrals==
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| {{For|a full list|list of integrals of hyperbolic functions}}
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| <math>\begin{align}
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| \int \sinh (ax)\,dx &= a^{-1} \cosh (ax) + C \\
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| \int \cosh (ax)\,dx &= a^{-1} \sinh (ax) + C \\
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| \int \tanh (ax)\,dx &= a^{-1} \ln (\cosh (ax)) + C \\
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| \int \coth (ax)\,dx &= a^{-1} \ln (\sinh (ax)) + C \\
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| \int \operatorname{sech} (ax)\,dx &= a^{-1} \arctan (\sinh (ax)) + C \\
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| \int \operatorname{csch} (ax)\,dx &= a^{-1} \ln \left( \tanh \left( \frac{ax}{2} \right) \right) + C &= a^{-1} \ln\left|\operatorname{csch} (ax) - \coth (ax)\right| + C
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| \end{align}</math>
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| The following integrals can be proved using [[hyperbolic substitution]].
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| <math>\begin{align}
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| \int {\frac{du}{\sqrt{a^2 + u^2}}} & = \operatorname{arsinh} \left( \frac{u}{a} \right) + C \\
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| \int {\frac{du}{\sqrt{u^2 - a^2}}} &= \operatorname{arcosh} \left( \frac{u}{a} \right) + C \\
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| \int {\frac{du}{a^2 - u^2}} & = a^{-1}\operatorname{artanh} \left( \frac{u}{a} \right) + C; u^2 < a^2 \\
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| \int {\frac{du}{a^2 - u^2}} & = a^{-1}\operatorname{arcoth} \left( \frac{u}{a} \right) + C; u^2 > a^2 \\
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| \int {\frac{du}{u\sqrt{a^2 - u^2}}} & = -a^{-1}\operatorname{arsech}\left( \frac{u}{a} \right) + C \\
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| \int {\frac{du}{u\sqrt{a^2 + u^2}}} & = -a^{-1}\operatorname{arcsch}\left| \frac{u}{a} \right| + C
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| \end{align}</math>
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| where ''C'' is the [[constant of integration]].
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| ==Taylor series expressions==
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| It is possible to express the above functions as [[Taylor series]]:
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| :<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>
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| The function sinh ''x'' has a Taylor series expression with only odd exponents for ''x''. Thus it is an [[odd function]], that is, −sinh ''x'' = sinh(−''x''), and sinh 0 = 0.
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| :<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>
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| The function cosh ''x'' has a Taylor series expression with only even exponents for ''x''. Thus it is an [[even function]], that is, symmetric with respect to the ''y''-axis. The sum of the sinh and cosh series is the [[infinite series]] expression of the [[exponential function]].
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| :<math>\begin{align}
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| \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)!}, \left |x \right | < \frac {\pi} {2} \\
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| \coth x &= x^{-1} + \frac {x} {3} - \frac {x^3} {45} + \frac {2x^5} {945} + \cdots = x^{-1} + \sum_{n=1}^\infty \frac{2^{2n} B_{2n} x^{2n-1}} {(2n)!}, 0 < \left |x \right | < \pi \\
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| \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)!} , \left |x \right | < \frac {\pi} {2} \\
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| \operatorname {csch}\, x &= x^{-1} - \frac {x} {6} +\frac {7x^3} {360} -\frac {31x^5} {15120} + \cdots = x^{-1} + \sum_{n=1}^\infty \frac{ 2 (1-2^{2n-1}) B_{2n} x^{2n-1}}{(2n)!} , 0 < \left |x \right | < \pi
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| | |
| \end{align}</math>
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| | |
| where
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| :<math>B_n \,</math> is the ''n''th [[Bernoulli number]]
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| :<math>E_n \,</math> is the ''n''th [[Euler number]]
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| ==Comparison with circular functions==
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| [[File:Circular and hyperbolic angle.svg|right|250px|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''.]]
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| 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]].
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| Since the [[circular sector#Area|area of a circular sector]] is <math>\frac {r^2 u} {2} ,</math> it will be equal to ''u'' when ''r'' = [[square root of 2]]. In the diagram such a circle is tangent to the hyperbola ''x y'' = 1 at (1,1). The yellow sector depicts an area and angle magnitude. Similarly, the red augmentation depicts an area and magnitude as hyperbolic angle.
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| The legs of the two [[right triangle]]s with hypotenuse on the ray defining the angles are of length √2 times the circular and hyperbolic functions.
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| Mellon Haskell of [[University of California, Berkeley]] described the basis of hyperbolic functions in areas of [[hyperbolic sector]]s in an 1895 article in [[Bulletin of the American Mathematical Society]] (see External links). He refers to the hyperbolic angle as an [[invariant measure]] with respect to the [[squeeze mapping]] just as circular angle is invariant under rotation.
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| ==Identities==
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| The hyperbolic functions satisfy many identities, all of them similar in form to the [[trigonometric identity|trigonometric identities]]. In fact, '''Osborn's rule'''<ref>G. Osborn, [http://links.jstor.org/sici?sici=0025-5572(190207)2%3A2%3A34%3C189%3A1MFHF%3E2.0.CO%3B2-Z Mnemonic for hyperbolic formulae], The Mathematical Gazette, p. 189, volume 2, issue 34, July 1902</ref> 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 2, 6, 10, 14, ... sinhs. This yields for example the addition theorems
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| :<math>\begin{align}
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| \sinh(x + y) &= \sinh (x) \cosh (y) + \cosh (x) \sinh (y) \\
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| \cosh(x + y) &= \cosh (x) \cosh (y) + \sinh (x) \sinh (y) \\
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| \tanh(x + y) &= \frac{\tanh (x) + \tanh (y)}{1 + \tanh (x) \tanh (y)}
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| \end{align}</math>
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| | |
| the "double argument formulas"
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| :<math>\begin{align}
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| \sinh 2x &= 2\sinh x \cosh x \\
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| \cosh 2x &= \cosh^2 x + \sinh^2 x = 2\cosh^2 x - 1 = 2\sinh^2 x + 1 \\
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| \tanh 2x &= \frac{2\tanh x}{1 + \tanh^2 x}\\
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| \sinh 2x &= \frac{2\tanh x}{1-\tanh^2 x}\\
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| \cosh 2x &= \frac{1+ \tanh^2 x}{1-\tanh^2 x}
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| \end{align}</math>
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| | |
| and the "half-argument formulas"<ref>
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| {{cite book
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| |title=Technical mathematics with calculus
| |
| |edition=3rd
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| |first1=John Charles
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| |last1=Peterson
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| |publisher=Cengage Learning
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| |year=2003
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| |isbn=0-7668-6189-9
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| |page=1155
| |
| |url=http://books.google.com/books?id=PGuSDjHvircC}}, [http://books.google.com/books?id=PGuSDjHvircC&pg=PA1155 Chapter 26, page 1155]
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| </ref>
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| :<math>\sinh \frac{x}{2} = \sqrt{ \frac{1}{2}(\cosh x - 1)} \,</math> Note: This is equivalent to its circular counterpart multiplied by −1.
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| :<math>\cosh \frac{x}{2} = \sqrt{ \frac{1}{2}(\cosh x + 1)} \,</math> Note: This corresponds to its circular counterpart.
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| :<math> \tanh \frac{x}{2} = \sqrt \frac{\cosh x - 1}{\cosh x + 1} = \frac{\sinh x}{\cosh x + 1} = \frac{\cosh x - 1}{\sinh x} = \coth x - \operatorname{csch}x.</math>
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| :<math> \coth \frac{x}{2} = \coth x + \operatorname{csch}x.</math>
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| The [[derivative]] of sinh ''x'' is cosh ''x'' and the derivative of cosh ''x'' is sinh ''x''; this is similar to trigonometric functions, albeit the sign is different (i.e., the derivative of cos ''x'' is −sin ''x'').
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| The [[Gudermannian function]] gives a direct relationship between the circular functions and the hyperbolic ones that does not involve complex numbers.
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| The graph of the function ''a'' cosh(''x''/''a'') is the [[catenary]], the curve formed by a uniform flexible chain hanging freely between two fixed points under uniform gravity.
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| ==Relationship to the exponential function==
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| From the definitions of the hyperbolic sine and cosine, we can derive the following identities:
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| :<math>e^x = \cosh x + \sinh x</math>
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| and
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| :<math>e^{-x} = \cosh x - \sinh x</math>
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| These expressions are analogous to the expressions for sine and cosine, based on [[Euler's formula]], as sums of complex exponentials.
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| ==Hyperbolic functions for complex numbers==
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| 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 function|holomorphic]].
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| Relationships to ordinary trigonometric functions are given by [[Euler's formula]] for complex numbers:
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| :<math>\begin{align}
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| e^{i x} &= \cos x + i \;\sin x \\
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| e^{-i x} &= \cos x - i \;\sin x
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| \end{align}</math>
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| so:
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| :<math>\begin{align}
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| \cosh ix &= \frac{1}{2} \left(e^{i x} + e^{-i x}\right) = \cos x \\
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| \sinh ix &= \frac{1}{2} \left(e^{i x} - e^{-i x}\right) = i \sin x \\
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| \cosh(x+iy) &= \cosh(x) \cos(y) + i \sinh(x) \sin(y) \\
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| \sinh(x+iy) &= \sinh(x) \cos(y) + i \cosh(x) \sin(y) \\
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| \tanh ix &= i \tan x \\
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| \cosh x &= \cos ix \\
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| \sinh x &= - i \sin ix \\
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| \tanh x &= - i \tan ix
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| \end{align}</math>
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| 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).
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| {| style="text-align:center"
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| |+ Hyperbolic functions in the complex plane
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| |[[Image:Complex Sinh.jpg|1000x140px|none]]
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| |[[Image:Complex Cosh.jpg|1000x140px|none]]
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| |[[Image:Complex Tanh.jpg|1000x140px|none]]
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| |[[Image:Complex Coth.jpg|1000x140px|none]]
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| |[[Image:Complex Sech.jpg|1000x140px|none]]
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| |[[Image:Complex Csch.jpg|1000x140px|none]]
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| |-
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| |<math>\operatorname{sinh}(z)</math>
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| |<math>\operatorname{cosh}(z)</math>
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| |<math>\operatorname{tanh}(z)</math>
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| |<math>\operatorname{coth}(z)</math>
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| |<math>\operatorname{sech}(z)</math>
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| |<math>\operatorname{csch}(z)</math>
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| |}
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| ==See also==
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| {{Commons category|Hyperbolic functions}}
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| * [[e (mathematical constant)]]
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| * [[Equal incircles theorem]], based on sinh
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| * [[Inverse hyperbolic function]]s
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| * [[List of integrals of hyperbolic functions]]
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| * [[Poinsot's spirals]]
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| * [[Sigmoid function]]
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| * [[Trigonometric functions]]
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| * [[Modified hyperbolic tangent]]
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| ==References==
| |
| {{Reflist}}
| |
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| ==External links==
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| * Mellon W. Haskell (1895) [http://www.ams.org/journals/bull/1895-01-06/S0002-9904-1895-00266-9/S0002-9904-1895-00266-9.pdf On the introduction of the notion of hyperbolic functions] [[Bulletin of the American Mathematical Society]] 1(6):155–9.
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| *{{springer|title=Hyperbolic functions|id=p/h048250}}
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| *[http://planetmath.org/hyperbolicfunctions Hyperbolic functions] on [[PlanetMath]]
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| *[http://mathworld.wolfram.com/HyperbolicFunctions.html Hyperbolic functions] entry at [[MathWorld]]
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| *[http://glab.trixon.se/ GonioLab]: Visualization of the unit circle, trigonometric and hyperbolic functions ([[Java Web Start]])
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| *[http://www.calctool.org/CALC/math/trigonometry/hyperbolic Web-based calculator of hyperbolic functions]
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| {{DEFAULTSORT:Hyperbolic Function}}
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| [[Category:Elementary special functions]]
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| [[Category:Exponentials]]
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| [[Category:Hyperbolic geometry]]
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| [[Category:Analytic functions]]
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