Hyperbolic functions

{{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, '''hyperbolic functions''' are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points {{math|(cos ''t'', sin ''t'')}} form a 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.

Hyperbolic functions are used to express the angle of parallelism in hyperbolic geometry. They are used to express Lorentz boosts as hyperbolic rotations in special relativity. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, and 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 are: * '''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> * '''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 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 animated version with comparison with the trigonometric (circular) functions).]]

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

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 functions. As a result, the other hyperbolic functions are meromorphic in the whole complex plane.

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

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

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

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

Hyperbolic functions were formally introduced in 1757 by Vincenzo Riccati.<ref name=":0" /><ref name=":3" /><ref name=":4" /> Riccati used {{math|''Sc.''}} and {{math|''Cc.''}} ({{lang|la|sinus/cosinus circulare}}) to refer to circular functions and {{math|''Sh.''}} and {{math|''Ch.''}} ({{lang|la|sinus/cosinus hyperbolico}}) to refer to hyperbolic functions.<ref name=":0" /> As early as 1759, Daviet de Foncenex showed the interchangeability of the trigonometric and hyperbolic functions using the imaginary unit and extended de Moivre's formula to hyperbolic functions.<ref name=":4" /><ref name=":0" />

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

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

==Definitions== thumb|right|250px|Right triangles with legs proportional to sinh and cosh With hyperbolic angle ''u'', the hyperbolic functions sinh and cosh can be defined with the exponential function eu.<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 of {{math|''ex''}} and {{math|''e''−''x''}}]] [[File:Hyperbolic and exponential; cosh.svg|thumb|right|{{math|cosh ''x''}} is the average of {{math|''ex''}} and {{math|''e''−''x''}}]]

* Hyperbolic sine: the odd part of the exponential function, that is, <math display="block"> \sinh x = \frac {e^x - e^{-x}} {2} = \frac {e^{2x} - 1} {2e^x}.</math> * Hyperbolic cosine: the even part of the exponential function, that is, <math display="block"> \cosh x = \frac {e^x + e^{-x}} {2} = \frac {e^{2x} + 1} {2e^x}.</math> thumb|sinh, cosh and tanh thumb|csch, sech and coth * 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 equations: 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'' ″(''x'') = ''f'' (''x'')}}, such that {{math|1=''f'' (0) = 1}}, {{math|1=''f'' ′(0) = 0}} for the hyperbolic cosine, and {{math|1=''f'' (0) = 0}}, {{math|1=''f'' ′(0) = 1}} for the hyperbolic sine.

=== Complex trigonometric definitions ===

Hyperbolic functions may also be deduced from trigonometric functions with complex arguments:

* Hyperbolic sine:<ref name=":1" /> <math display="block">\sinh x = -i \sin (i x).</math> * Hyperbolic cosine:<ref name=":1" /> <math display="block">\cosh x = \cos (i x).</math> * Hyperbolic tangent: <math display="block">\tanh x = -i \tan (i x).</math> * Hyperbolic cotangent: <math display="block">\coth x = i \cot (i x).</math> * Hyperbolic secant: <math display="block"> \operatorname{sech} x = \sec (i x).</math> * Hyperbolic cosecant:<math display="block">\operatorname{csch} x = i \csc (i x).</math> where {{mvar|i}} is the imaginary unit 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).

== Characterizing properties==

=== Hyperbolic cosine ===

It can be shown that the area under the curve of the hyperbolic cosine (over a finite interval) is always equal to the arc length 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 {{math|1=''f'' ′ = 1 − ''f'' 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>

==Useful relations== {{Anchor|Osborn}} The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, '''Osborn's rule'''<ref name="Osborn, 1902" /> (named after George Osborn) states that one can convert any trigonometric identity (up to but not including sinhs or implied sinhs of 4th degree) for <math>\theta</math>, <math>2\theta</math>, <math>3\theta</math> or <math>\theta</math> and <math>\varphi</math> into a hyperbolic identity, by: # expanding it completely in terms of integral powers of sines and cosines, # changing sine to sinh and cosine to cosh, and # switching the sign of every term containing a product of two sinhs.

Odd and even functions: <math display="block">\begin{align} \sinh (-x) &= -\sinh x \\ \cosh (-x) &= \cosh x \\ \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:

<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:

<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.

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>

==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 combinations of {{math|sinh}} and {{math|cosh}}, in particular the exponential functions <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 for every complex value of {{mvar|x}}. Since the function {{math|sinh ''x''}} is odd, only odd exponents for {{math|''x''}} occur in its Taylor series.

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

The sum of the sinh and cosh series is the infinite series expression of the exponential function.

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 {{math|(1, 1)}} display geometry of circular functions in terms of 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 functions. Both types depend on an argument, either circular angle or hyperbolic angle.

Since the 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 {{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 with area corresponding to hyperbolic angle magnitude.

The legs of the two right triangles with the 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>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 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 |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 can be defined for any 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.

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 with respect to the imaginary component, with period <math>2 \pi i</math> (<math>\pi i</math> for hyperbolic tangent and cotangent).

==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]

{{DEFAULTSORT:Hyperbolic Function}} Category:Hyperbolic functions Category:Exponentials Category:Hyperbolic geometry Category:Analytic functions Category:Sigmoid functions