# Hyperbolic functions

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{{Short description|Hyperbolic analogues of trigonometric functions}}
{{Redirect|Hyperbolic curve|the geometric curve|Hyperbola}}
{{Anchor|Sinh|Cosh|Tanh|Sech|Csch|Coth}}
333x333px|thumb

In [mathematics](/source/mathematics), '''hyperbolic functions''' are analogues of the ordinary [trigonometric function](/source/trigonometric_function)s, but defined using the [hyperbola](/source/hyperbola) rather than the [circle](/source/circle). Just as the points {{math|(cos ''t'', sin ''t'')}} form a [circle with a unit radius](/source/unit_circle), the points {{math|(cosh ''t'', sinh ''t'')}} form the right half of the [unit hyperbola](/source/unit_hyperbola). Also, similarly to how the derivatives of {{math|sin(''t'')}} and {{math|cos(''t'')}} are {{math|cos(''t'')}} and {{math|–sin(''t'')}} respectively, the derivatives of {{math|sinh(''t'')}} and {{math|cosh(''t'')}} are {{math|cosh(''t'')}} and {{math|sinh(''t'')}} respectively.

Hyperbolic functions are used to express the [angle of parallelism](/source/angle_of_parallelism) in [hyperbolic geometry](/source/hyperbolic_geometry). They are used to express [Lorentz boost](/source/Lorentz_boost)s as [hyperbolic rotation](/source/hyperbolic_rotation)s in [special relativity](/source/special_relativity). They also occur in the solutions of many linear [differential equation](/source/differential_equation)s (such as the equation defining a [catenary](/source/catenary)), [cubic equations](/source/Cubic_equation), and [Laplace's equation](/source/Laplace's_equation) in [Cartesian coordinates](/source/Cartesian_coordinates). [Laplace's equation](/source/Laplace's_equation)s are important in many areas of [physics](/source/physics), including [electromagnetic theory](/source/electromagnetic_theory), [heat transfer](/source/heat_transfer), and [fluid dynamics](/source/fluid_dynamics).

The basic hyperbolic functions are:<ref name=":1">{{Cite web|last=Weisstein|first=Eric W.|authorlink=Eric W. Weisstein|title=Hyperbolic Functions| url=https://mathworld.wolfram.com/HyperbolicFunctions.html|access-date=2020-08-29|website=mathworld.wolfram.com|language=en}}</ref>
* '''hyperbolic sine''' "{{math|sinh}}" ({{IPAc-en|ˈ|s|ɪ|ŋ|,_|ˈ|s|ɪ|n|tʃ|,_|ˈ|ʃ|aɪ|n}}),<ref>(1999) ''Collins Concise Dictionary'', 4th edition, HarperCollins, Glasgow, {{ISBN|0 00 472257 4}}, p. 1386</ref>
* '''hyperbolic cosine''' "{{math|cosh}}" ({{IPAc-en|ˈ|k|ɒ|ʃ|,_|ˈ|k|oʊ|ʃ}}),<ref name="Collins Concise Dictionary p. 328">''Collins Concise Dictionary'', p. 328</ref>
from which are derived:<ref name=":2">{{Cite web|title=Hyperbolic Functions|url=https://www.mathsisfun.com/sets/function-hyperbolic.html|access-date=2020-08-29|website=www.mathsisfun.com}}</ref>
* '''hyperbolic tangent''' "{{math|tanh}}" ({{IPAc-en|ˈ|t|æ|ŋ|,_|ˈ|t|æ|n|tʃ|,_|ˈ|θ|æ|n}}),<ref>''Collins Concise Dictionary'', p. 1520</ref>
* '''hyperbolic cotangent''' "{{math|coth}}" ({{IPAc-en|ˈ|k|ɒ|θ|,_|ˈ|k|oʊ|θ}}),<ref>''Collins Concise Dictionary'', p. 329</ref><ref>[http://www.mathcentre.ac.uk/resources/workbooks/mathcentre/hyperbolicfunctions.pdf tanh]</ref>
* '''hyperbolic secant''' "{{math|sech}}" ({{IPAc-en|ˈ|s|ɛ|tʃ|,_|ˈ|ʃ|ɛ|k}}),<ref>''Collins Concise Dictionary'', p. 1340</ref>
* '''hyperbolic cosecant''' "{{math|csch}}" or "{{math|cosech}}" ({{IPAc-en|ˈ|k|oʊ|s|ɛ|tʃ|,_|ˈ|k|oʊ|ʃ|ɛ|k}}<ref name="Collins Concise Dictionary p. 328"/>)
corresponding to the derived trigonometric functions.

The [inverse hyperbolic functions](/source/inverse_hyperbolic_functions) are:
* '''inverse hyperbolic sine''' "{{math|arsinh}}" (also denoted "{{math|sinh<sup>−1</sup>}}", "{{math|asinh}}" or sometimes "{{math|arcsinh}}")<ref>{{Citation | last=Woodhouse | first = N. M. J. | author-link = N. M. J. Woodhouse | title = Special Relativity | publisher = Springer | place = London | date = 2003 | page = 71 | isbn = 978-1-85233-426-0}}</ref><ref>{{Citation | editor1-last=Abramowitz | editor1-first=Milton | editor1-link=Milton Abramowitz | editor2-last=Stegun | editor2-first=Irene A. | editor2-link=Irene Stegun | title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables | publisher=[Dover Publications](/source/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](/source/Google_Books).</ref>
* '''inverse hyperbolic cosine''' "{{math|arcosh}}" (also denoted "{{math|cosh<sup>−1</sup>}}", "{{math|acosh}}" or sometimes "{{math|arccosh}}")
* '''inverse hyperbolic tangent''' "{{math|artanh}}" (also denoted "{{math|tanh<sup>−1</sup>}}", "{{math|atanh}}" or sometimes "{{math|arctanh}}")
* '''inverse hyperbolic cotangent''' "{{math|arcoth}}" (also denoted "{{math|coth<sup>−1</sup>}}", "{{math|acoth}}" or sometimes "{{math|arccoth}}")
* '''inverse hyperbolic secant''' "{{math|arsech}}" (also denoted "{{math|sech<sup>−1</sup>}}", "{{math|asech}}" or sometimes "{{math|arcsech}}")
* '''inverse hyperbolic cosecant''' "{{math|arcsch}}" (also denoted "{{math|arcosech}}", "{{math|csch<sup>−1</sup>}}", "{{math|cosech<sup>−1</sup>}}","{{math|acsch}}", "{{math|acosech}}", or sometimes "{{math|arccsch}}" or "{{math|arccosech}}")
[[File:Hyperbolic functions-2.svg|thumb|upright=1.4|A [ray](/source/Ray_(geometry)) through the [unit hyperbola](/source/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](/source/%3AImage%3AHyperbolicAnimation.gif) with comparison with the trigonometric (circular) functions).]]

The hyperbolic functions take an [argument](/source/argument_of_a_function) called a [hyperbolic angle](/source/hyperbolic_angle). The magnitude of a hyperbolic angle is the [area](/source/area) of its [hyperbolic sector](/source/hyperbolic_sector) to {{math|1=''xy'' = 1}}. The hyperbolic functions may be defined in terms of the [legs of a right triangle](/source/hyperbolic_sector) covering this sector.

In [complex analysis](/source/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](/source/entire_function)s. As a result, the other hyperbolic functions are [meromorphic](/source/meromorphic_function) in the whole complex plane.

By [Lindemann–Weierstrass theorem](/source/Lindemann%E2%80%93Weierstrass_theorem), the hyperbolic functions have a [transcendental value](/source/transcendental_number) for every non-zero [algebraic value](/source/algebraic_number) of the argument.<ref>{{Cite book | jstor=10.4169/j.ctt5hh8zn| title=Irrational Numbers | volume=11| last1=Niven| first1=Ivan| year=1985| publisher=Mathematical Association of America| isbn=9780883850381}}</ref>

== History ==
The first known calculation of a hyperbolic trigonometry problem is attributed to [Gerardus Mercator](/source/Gerardus_Mercator) when issuing the [Mercator map projection](/source/Mercator_projection) circa 1566. It requires tabulating solutions to a [transcendental equation](/source/transcendental_equation) involving hyperbolic functions.<ref name=":3">{{Cite book |last=George F. Becker |url=https://archive.org/details/hyperbolicfuncti027682mbp/page/n49/mode/2up?q=mercator |title=Hyperbolic Functions |last2=C. E. Van Orstrand |date=1909 |publisher=The Smithsonian Institution |others=Universal Digital Library}}</ref>

The first to suggest a similarity between the sector of the circle and that of the hyperbola was [Isaac Newton](/source/Isaac_Newton) in his 1687 [''Principia Mathematica''](/source/Philosophi%C3%A6_Naturalis_Principia_Mathematica).<ref name=":0">{{Cite book |last=McMahon |first=James |url=https://archive.org/details/hyperbolicfuncti031883mbp/page/n75/mode/2up |title=Hyperbolic Functions |date=1896 |publisher=John Wiley And Sons |others=Osmania University, Digital Library Of India}}</ref>

[Roger Cotes](/source/Roger_Cotes) suggested to modify the trigonometric functions using the [imaginary unit](/source/imaginary_unit) <math>i=\sqrt{-1} </math> to obtain an oblate [spheroid](/source/spheroid) from a prolate one.<ref name=":0" />

Hyperbolic functions were formally introduced in 1757 by [Vincenzo Riccati](/source/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, [Daviet de Foncenex](/source/Fran%C3%A7ois_Daviet_de_Foncenex) showed the interchangeability of the trigonometric and hyperbolic functions using the imaginary unit and extended [de Moivre's formula](/source/de_Moivre's_formula) to hyperbolic functions.<ref name=":4" /><ref name=":0" />

During the 1760s, [Johann Heinrich Lambert](/source/Johann_Heinrich_Lambert) systematized the use functions and provided exponential expressions in various publications.<ref name=":0" /><ref name=":4">Bradley, Robert E.; D'Antonio, Lawrence A.; Sandifer, Charles Edward. ''Euler at 300: an appreciation.'' Mathematical Association of America, 2007. Page 100.</ref> Lambert credited Riccati for the terminology and names of the functions, but altered the abbreviations to those used today.<ref name=":4" /><ref>Becker, Georg F. ''Hyperbolic functions.'' Read Books, 1931. Page xlviii.</ref>

== Notation ==
{{main|Trigonometric functions#Notation}}

==Definitions==
thumb|right|250px|Right triangles with legs proportional to sinh and cosh
With [hyperbolic angle](/source/hyperbolic_angle) ''u'', the hyperbolic functions sinh and cosh can be defined with the [exponential function](/source/exponential_function) e<sup>u</sup>.<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 ===
[[File:Hyperbolic and exponential; sinh.svg|thumb|right|{{math|sinh ''x''}} is half the [difference](/source/Subtraction) 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 [average](/source/Arithmetic_mean) of {{math|''e<sup>x</sup>''}} and {{math|''e''<sup>−''x''</sup>}}]]

* Hyperbolic sine: the [odd part](/source/odd_part_of_a_function) of the exponential function, that is, <math display="block"> \sinh x = \frac {e^x - e^{-x}} {2} = \frac {e^{2x} - 1} {2e^x}.</math>
* Hyperbolic cosine: the [even part](/source/even_part_of_a_function) of the exponential function, that is, <math display="block"> \cosh x = \frac {e^x + e^{-x}} {2} = \frac {e^{2x} + 1} {2e^x}.</math>
thumb|<span style="color:#b30000;">sinh</span>, <span style="color:#00b300;">cosh</span> and <span style="color:#0000b3;">tanh</span>
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 ===

The hyperbolic functions may be defined as solutions of [differential equation](/source/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 function](/source/trigonometric_function)s with [complex](/source/complex_number) 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](/source/imaginary_unit) with {{math|1=''i''<sup>2</sup> = −1}}.

The above definitions are related to the exponential definitions via [Euler's formula](/source/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](/source/area_under_the_curve) of the hyperbolic cosine (over a finite interval) is always equal to the [arc length](/source/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](/source/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 |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>

==Useful relations==
{{Anchor|Osborn}}
The hyperbolic functions satisfy many identities, all of them similar in form to the [trigonometric identities](/source/trigonometric_identity). In fact, '''Osborn's rule'''<ref name="Osborn, 1902" /> (named after [George Osborn](/source/George_Osborn_(mathematician))) 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](/source/Odd_function) and [even](/source/Even_function) functions:
<math display="block">\begin{align}
               \sinh (-x) &= -\sinh x \\
               \cosh (-x) &=  \cosh x \\
               \tanh (-x) &= -\tanh x \\
               \coth (-x) &= -\coth x \\
 \operatorname{sech} (-x) &=  \operatorname{sech} x \\
 \operatorname{csch} (-x) &= -\operatorname{csch} x
\end{align}</math>

Reciprocals:

<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>

Analogous to [Euler's formula](/source/Euler's_formula):

<math display="block">\begin{align}
 \cosh x + \sinh x &= e^x \\
 \cosh x - \sinh x &= e^{-x} 
\end{align}</math>

Analogous to the [Pythagorean trigonometric identity](/source/Pythagorean_trigonometric_identity):

<math display="block">\begin{align}
 \cosh^2 x - \sinh^2 x &= 1 \\
 1 - \tanh^{2} x &= \operatorname{sech} ^{2} x  \\
 \coth^{2} x - 1 &= \operatorname{csch} ^{2} x
\end{align}</math>

===Sums and differences 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 } \\
 \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>

===Addition and subtraction formulas===
<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)\\
 \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>

===Product formulas===
<math display=block>\begin{align}
\cosh x\, \cosh y &= \tfrac12\bigl(\!\!~\cosh(x + y) + \cosh(x - y)\bigr) \\[5mu]
\sinh x\, \sinh y &= \tfrac12\bigl(\!\!~\cosh(x + y) - \cosh(x - y)\bigr) \\[5mu]
\sinh x\, \cosh y &= \tfrac12\bigl(\!\!~\sinh(x + y) + \sinh(x - y)\bigr) \\[5mu]
\cosh x\, \sinh y &= \tfrac12\bigl(\!\!~\sinh(x + y) - \sinh(x - y)\bigr) \\[5mu]
\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](/source/sign_function).

If {{math|''x'' ≠ 0}} then

<math display="block"> \tanh\left(\frac{x}{2}\right) = \frac{\cosh x - 1}{\sinh x} = \coth x - \operatorname{csch} x </math>

===Tangent half argument formulas===
When {{tmath|1=t = \tanh\left(\frac{x}{2}\right) }},
<math display="block">
\begin{align}
& \sinh x = \frac{2t}{1 - t^2},
& & \cosh x = \frac{1 + t^2}{1 - t^2}, \\[8pt]
& \tanh x = \frac{2t}{1 + t^2},
& & \coth x = \frac{1 + t^2}{2t}, \\[8pt]
& \operatorname{sech} x = \frac{1 - t^2}{1 + t^2},
& & \operatorname{csch} x = \frac{1 - t^2}{2t}.
\end{align}
</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:<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 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.

==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](/source/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](/source/linear_combination)s of {{math|sinh}} and {{math|cosh}}, in particular the [exponential function](/source/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](/source/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](/source/constant_of_integration).

==Taylor series expressions==
It is possible to express explicitly the [Taylor series](/source/Taylor_series) at zero (or the [Laurent series](/source/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](/source/convergent_series) for every [complex](/source/complex_number) value of {{mvar|x}}. Since the function {{math|sinh ''x''}} is [odd](/source/odd_function), 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](/source/convergent_series) for every [complex](/source/complex_number) value of {{mvar|x}}. Since the function {{math|cosh ''x''}} is [even](/source/even_function), only even exponents for {{mvar|x}} occur in its Taylor series.

The sum of the sinh and cosh series is the [infinite series](/source/infinite_series) expression of the [exponential function](/source/exponential_function).

The following series are followed by a description of a subset of their [domain of convergence](/source/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](/source/Bernoulli_number)
*<math>E_n </math> is the ''n''th [Euler number](/source/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 {{math|(1, 1)}} display geometry of circular functions in terms of [circular sector](/source/sector_of_a_circle) area {{mvar|u}} and hyperbolic functions depending on [hyperbolic sector](/source/hyperbolic_sector) area {{mvar|u}}.]]
The hyperbolic functions represent an expansion of [trigonometry](/source/trigonometry) beyond the [circular function](/source/circular_function)s. Both types depend on an [argument](/source/argument_of_a_function), either [circular angle](/source/angle) or [hyperbolic angle](/source/hyperbolic_angle).

Since the [area of a circular sector](/source/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 {{math|1=''xy'' = 1}} at {{math|(1, 1)}}. The yellow sector depicts an area and angle magnitude. Similarly, the yellow and red regions together depict a [hyperbolic sector](/source/hyperbolic_sector) with area corresponding to hyperbolic angle magnitude.

The legs of the two [right triangle](/source/right_triangle)s with the [hypotenuse](/source/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](/source/invariant_measure) with respect to the [squeeze mapping](/source/squeeze_mapping), just as the circular angle is invariant under rotation.<ref>[Haskell, Mellen W.](/source/Mellen_W._Haskell), "On the introduction of the notion of hyperbolic functions",  [Bulletin of the American Mathematical Society](/source/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](/source/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](/source/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](/source/even%E2%80%93odd_decomposition) 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](/source/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](/source/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
|1000x140px|none
|1000x140px|none
|1000x140px|none
|1000x140px|none
|1000x140px|none
|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](/source/exponential_function) can be defined for any [complex](/source/complex_number) 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](/source/Holomorphic_function).

Relationships to ordinary trigonometric functions are given by [Euler's formula](/source/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](/source/periodic_function) 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)](/source/e_(mathematical_constant))
* [Equal incircles theorem](/source/Equal_incircles_theorem), based on sinh
* [Hyperbolastic functions](/source/Hyperbolastic_functions)
* [Hyperbolic growth](/source/Hyperbolic_growth)
* [Inverse hyperbolic function](/source/Inverse_hyperbolic_function)s
* [List of integrals of hyperbolic functions](/source/List_of_integrals_of_hyperbolic_functions)
* [Poinsot's spirals](/source/Poinsot's_spirals)
* [Sigmoid function](/source/Sigmoid_function)
* [Trigonometric functions](/source/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](/source/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](/source/PlanetMath)
*[https://web.archive.org/web/20071006172054/http://glab.trixon.se/ GonioLab]: Visualization of the unit circle, trigonometric and hyperbolic functions ([Java Web Start](/source/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}}
Category:Hyperbolic functions
Category:Exponentials
Category:Hyperbolic geometry
Category:Analytic functions
Category:Sigmoid functions

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Adapted from the Wikipedia article [Hyperbolic functions](https://en.wikipedia.org/wiki/Hyperbolic_functions) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Hyperbolic_functions?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.