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[[Berkas:Hyperbolic functions-2.svg|jmpl|ka|200px|Fungsi hiperbolik]]
'''Fungsi hiperbolik''' adalah salah satu hasil [[kombinasi]] dari fungsi-fungsi [[eksponen]]. Fungsi hiperbolik memiliki rumus. Selain itu memiliki [[invers]] serta turunan dan anti turunan fungsi hiperbolik dan inversnya.<ref name="Dwi Perpus Unnes">{{cite web|date=|title=FUNGSI HIPERBOLIK DAN INVERSNYA|url=http://lib.unnes.ac.id/990/|work=|publisher=DIGILIB UNNES|format=|doi=|accessdate=2014-05-28|archive-date=2019-08-15|archive-url=https://web.archive.org/web/20190815042811/https://lib.unnes.ac.id/990/|dead-url=no}}</ref>
==
[[Berkas:sinh cosh tanh.svg|thumb|<span style="color:#b30000;">sinh</span>, <span style="color:#00b300;">cosh</span> dan <span style="color:#0000b3;">tanh</span>]]
[[Berkas:csch sech coth.svg|thumb|<span style="color:#b30000;">csch</span>, <span style="color:#00b300;">sech</span> dan <span style="color:#0000b3;">coth</span>]]
=== Definisi Eksponen ===
[[Berkas:Hyperbolic and exponential; sinh.svg|thumb|right|{{math|sinh ''x''}} adalah separuh [[Pengurangan|selisih]] {{math|''e<sup>x</sup>''}} dan {{math|''e''<sup>−''x''</sup>}}]]
[[File:Hyperbolic and exponential; cosh.svg|thumb|right|{{math|cosh ''x''}} adalah [[Rata-rata aritmetika|rerata]] {{math|''e<sup>x</sup>''}} dan {{math|''e''<sup>−''x''</sup>}}]]
Dalam istilah dari [[fungsi eksponensial]]:
* Hiperbolik sinus:
*:<math>\sinh x = \frac {e^x - e^{-x}} {2} = \frac {e^{2x} - 1} {2e^x} = \frac {1 - e^{-2x}} {2e^{-x}}.</math>
* Hiperbolik kosinus:
*:<math>\cosh x = \frac {e^x + e^{-x}} {2} = \frac {e^{2x} + 1} {2e^x} = \frac {1 + e^{-2x}} {2e^{-x}}.</math>
* Hiperbolik tangen:
*:<math>\tanh x = \frac{\sinh x}{\cosh x} = \frac {e^x - e^{-x}} {e^x + e^{-x}}
= \frac{e^{2x} - 1} {e^{2x} + 1}</math>
* Hiperbolik kotangen: untuk {{math|''x'' ≠ 0}},
*:<math>\coth x = \frac{\cosh x}{\sinh x} = \frac {e^x + e^{-x}} {e^x - e^{-x}}
= \frac{e^{2x} + 1} {e^{2x} - 1}</math>
* Hiperbolik sekan:
*:<math>\operatorname{sech} x = \frac{1}{\cosh x} = \frac {2} {e^x + e^{-x}}
= \frac{2e^x} {e^{2x} + 1}</math>
* Hiperbolik kosekan: untuk {{math|''x'' ≠ 0}},
*:<math>\operatorname{csch} x = \frac{1}{\sinh x} = \frac {2} {e^x - e^{-x}}
= \frac{2e^x} {e^{2x} - 1}</math>
=== Definisi persamaan diferensial ===
- Dalam pengembangan -
<!--The hyperbolic functions may be defined as solutions of [[differential equation]]s: The hyperbolic sine and cosine are the unique solution {{math|(''s'', ''c'')}} of the system
:<math>\begin{align}
c'(x)&=s(x)\\
s'(x)&=c(x)\end{align}</math>
such that
{{math|1=''s''(0) = 0}} and {{math|1=''c''(0) = 1}}.
They 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.-->
=== Definisi kompleks trigonometri ===
-Dalam pengembangan -
<!--Hyperbolic functions may also be deduced from [[trigonometric function]]s with [[complex number|complex]] arguments:
* Hyperbolic sine:
*:<math>\sinh x = -i \sin (i x)</math>
* Hyperbolic cosine:
*:<math>\cosh x = \cos (i x)</math>
* Hyperbolic tangent:
*:<math>\tanh x = -i \tan (i x)</math>
* Hyperbolic cotangent:
*:<math>\coth x = i \cot (i x)</math>
* Hyperbolic secant:
*:<math>\operatorname{sech} x = \sec (i x)</math>
* Hyperbolic cosecant:
*:<math>\operatorname{csch} x = i \csc (i x)</math>
where {{mvar|i}} is the [[imaginary unit]] with {{math|1=''i''<sup>2</sup> = −1}}.
The above definitions are related to the exponential definitions via [[Euler's formula]]. (See "Hyperbolic functions for complex numbers" below.)-->
== Sifat karakteristik ==
- Dalam pengembangan -
<!--=== 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>\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>
{{Anchor|Tanh}}
===Hyperbolic tangent ===
The hyperbolic tangent is the solution to the [[differential equation]] {{math|1=''f'' ′ = 1 − ''f'' <sup>2</sup>}} with {{math|1=''f'' (0) = 0}} and the [[nonlinear]] [[boundary value problem]]:<ref>{{MathWorld | file = HyperbolicTangent | title = Hyperbolic Tangent}}</ref><ref>
{{cite web|title=Derivation of tanh solution to {{sfrac|1|2}}''f''" = ''f''<sup>3</sup> − ''f''|url=https://math.stackexchange.com/q/1670143 |website=Math [[StackExchange]]|accessdate=18 March 2016}}</ref>
:<math>\tfrac{1}{2} f'' = f^3 - f ; \quad f(0) = f'(\infty) = 0.</math>
<!--==Useful relations==
The hyperbolic functions satisfy many identities, all of them similar in form to the [[trigonometric identity|trigonometric identities]]. In fact, '''Osborn's rule'''<ref name="Osborn, 1902" /> states that one can convert any trigonometric identity 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 which contains a product of two sinhs.
Odd and even functions:
:<math>\begin{align}
\sinh (-x) &= -\sinh x \\
\cosh (-x) &= \cosh x
\end{align}</math>
Hence:
:<math>\begin{align}
\tanh (-x) &= -\tanh x \\
\coth (-x) &= -\coth x \\
\operatorname{sech} (-x) &= \operatorname{sech} x \\
\operatorname{csch} (-x) &= -\operatorname{csch} x
\end{align}</math>
Thus, {{math|cosh ''x''}} and {{math|sech ''x''}} are [[even function]]s; the others are [[odd functions]].
:<math>\begin{align}
\operatorname{arsech} x &= \operatorname{arcosh} \left(\frac{1}{x}\right) \\
\operatorname{arcsch} x &= \operatorname{arsinh} \left(\frac{1}{x}\right) \\
\operatorname{arcoth} x &= \operatorname{artanh} \left(\frac{1}{x}\right)
\end{align}</math>
Hyperbolic sine and cosine satisfy:
:<math>\begin{align}
\cosh x + \sinh x &= e^x \\
\cosh x - \sinh x &= e^{-x} \\
\cosh^2 x - \sinh^2 x &= 1
\end{align}</math>
the last of which is similar to the [[Pythagorean trigonometric identity]].
One also has
:<math>\begin{align}
\operatorname{sech} ^{2} x &= 1 - \tanh^{2} x \\
\operatorname{csch} ^{2} x &= \coth^{2} x - 1
\end{align}</math>
for the other functions.-->
=== Penambahan ===
:<math>\begin{align}
\sinh(x + y) &= \sinh x \cosh y + \cosh x \sinh y \\
\cosh(x + y) &= \cosh x \cosh y + \sinh x \sinh y \\[6px]
\tanh(x + y) &= \frac{\tanh x +\tanh y}{1+ \tanh x \tanh y } \\
\end{align}</math>
terutama
:<math>\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>
Lihat:
:<math>\begin{align}
\sinh x + \sinh y &= 2 \sinh \left(\frac{x+y}{2}\right) \cosh \left(\frac{x-y}{2}\right)\\
\cosh x + \cosh y &= 2 \cosh \left(\frac{x+y}{2}\right) \cosh \left(\frac{x-y}{2}\right)\\
\end{align}</math>
=== Pengurangan ===
:<math>\begin{align}
\sinh(x - y) &= \sinh x \cosh y - \cosh x \sinh y \\
\cosh(x - y) &= \cosh x \cosh y - \sinh x \sinh y \\
\tanh(x - y) &= \frac{\tanh x -\tanh y}{1- \tanh x \tanh y } \\
\end{align}</math>
Dan juga:<ref>{{cite book|last1=Martin|first1=George E.|title=The foundations of geometry and the non-euclidean plane|date=1986|publisher=Springer-Verlag|location=New York|isbn=3-540-90694-0|page=416|edition=1st corr.}}</ref>
:<math>\begin{align}
\sinh x - \sinh y &= 2 \cosh \left(\frac{x+y}{2}\right) \sinh \left(\frac{x-y}{2}\right)\\
\cosh x - \cosh y &= 2 \sinh \left(\frac{x+y}{2}\right) \sinh \left(\frac{x-y}{2}\right)\\
\end{align}</math>
=== Rumus setengah argumen ===
:<math>\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>
di mana {{math|sgn}} adalah [[fungsi tanda]].
Jika <math>x \ne 0</math>, maka<ref>{{cite web |title=Prove the identity |url=https://math.stackexchange.com/q/1565753 |website=[[StackExchange]] (mathematics) |accessdate=24 January 2016 |archive-date=2023-07-26 |archive-url=https://web.archive.org/web/20230726134114/https://math.stackexchange.com/questions/1565753/prove-the-identity-tanh-left-fracx2-right-frac-coshx-1-sinhx |dead-url=no }}</ref>
:<math> \tanh\left(\frac{x}{2}\right) = \frac{\cosh x - 1}{\sinh x} = \coth x - \operatorname{csch} x </math>
=== Rumus kuadrat ===
:<math>\begin{align}
\sinh^2 x &= \frac{1}{2}(\cosh 2x -1) \\
\cosh^2 x &= \frac{1}{2}(\cosh 2x +1)
\end{align}</math>
=== Pertidaksamaan ===
Pertidaksamaan berikut sangat berguna dalam statistik, yaitu <math>\operatorname{cosh}(t) \leq e^{t^2 /2}</math><ref>{{cite article|last1=Audibert|first1=Jean-Yves|title=Fast learning rates in statistical inference through aggregation|date=2009|publisher=The Annals of Statistics|page=1627}} [https://projecteuclid.org/download/pdfview_1/euclid.aos/1245332827] {{Webarchive|url=https://web.archive.org/web/20230726134112/https://projecteuclid.org/journalArticle/Download?urlId=10.1214%2F08-AOS623&isResultClick=False |date=2023-07-26 }}</ref>
== Fungsi invers sebagai logaritma ==
{{main|Fungsi hiperbolik invers}}
:<math>\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 \geqslant 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 \leqslant 1 \\
\operatorname {arcsch} (x) &= \ln \left( \frac{1}{x} + \sqrt{\frac{1}{x^2} +1}\right) = \ln \left( \frac{1+ \sqrt{1 + x^2}}{x} \right) && x \ne 0
\end{align}</math>
<!--
Pembuktian <math>arcsinh x = ln (x + \sqrt{x^2 + 1})</math>!
: <math>y = arcsinh x</math>
: <math>x = arcsinh y</math>
: <math>x = \frac{e^y - e^{-y}}{2}</math>
: <math>2x = e^y - e^{-y}</math>
: <math>2e^yx = e^2y - 1</math>
: <math>(e^y)^2 - 2x(e^y) - 1 = 0</math>
: <math>e^y = \frac{2x + \sqrt{4x^2 + 4}}{2}</math>
: <math>e^y = x + \sqrt{x^2 + 1}</math>
: <math>ln e^y = ln (x + \sqrt{x^2 + 1})</math>
: <math>y ln e = ln (x + \sqrt{x^2 + 1})</math>
: <math>y = ln (x + \sqrt{x^2 + 1})</math>
: <math>arcsinh x = ln (x + \sqrt{x^2 + 1})</math>
-->
== Turunan ==
:<math>\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 \\
\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>
<!-- 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 -->
== Turunan detik ==
- Dalam pengembangan -
<!--Sinh and cosh are both equal to their [[second derivative]], that is:
:<math> \frac{d^2}{dx^2}\sinh x = \sinh x \,</math>
:<math> \frac{d^2}{dx^2}\cosh x = \cosh x \, .</math>
<!--All functions with this property are [[linear combination]]s of sinh and cosh, in particular the [[exponential function]]s <math> e^x </math> and <math> e^{-x} </math>.-->
== Standar integral ==
<!--{{For|a full list|list of integrals of hyperbolic functions}}-->
:<math>\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 (\sinh (ax)) + 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|\operatorname{csch} (ax) - \coth (ax)\right| + C
\end{align}</math>
----
:<math>\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} &= \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>
<!--==Taylor series expressions==
It is possible to express the above functions as [[Taylor series]]:
:<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>
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.
:<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>
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]].
:<math>\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)!}, \left |x \right | < \frac {\pi} {2} \\
\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 \\
\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} \\
\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
\end{align}</math>
where:
:<math>B_n \,</math> is the ''n''th [[Bernoulli number]]
:<math>E_n \,</math> is the ''n''th [[Euler number]]
==Comparison with circular functions==
[[File:Circular and hyperbolic angle.svg|right|upright=1.2|thumb|Circle and hyperbola tangent at (1,1) display geometry of circular functions in terms of [[sector of a circle|circular sector]] area ''u'' and hyperbolic functions depending on [[hyperbolic sector]] area ''u''.]]
The hyperbolic functions represent an expansion of [[trigonometry]] beyond the [[circular function]]s. Both types depend on an [[argument of a function|argument]], either [[angle|circular angle]] or [[hyperbolic angle]].
Since the [[circular sector#Area|area of a circular sector]] with radius ''r'' and angle ''u'' (in radians) is ''r''<sup>2</sup>''u''/2, it will be equal to ''u'' when ''r'' = {{sqrt|2}}. In the diagram such a circle is tangent to the hyperbola ''xy'' = 1 at (1,1). The yellow sector depicts an area and angle magnitude. Similarly, the yellow and red sectors together depict an area and [[hyperbolic sector|hyperbolic angle magnitude]].
The legs of the two [[right triangle]]s with hypotenuse on the ray defining the angles are of length {{radic|2}} times the circular and hyperbolic functions.
The hyperbolic angle is an [[invariant measure]] with respect to the [[squeeze mapping]], just as the circular angle is invariant under rotation.<ref>[[Mellen W. Haskell]], "On the introduction of the notion of hyperbolic functions", [[Bulletin of the American Mathematical Society]] '''1''':6:155–9, [http://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 ones that does not involve complex numbers.
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.
==Relationship to the exponential function==
The decomposition of the exponential function in its [[even–odd decomposition|even and odd parts]] gives the identities
:<math>e^x = \cosh x + \sinh x,</math>
and
:<math>e^{-x} = \cosh x - \sinh x.</math>
The first one is analogous to [[Euler's formula]]
:<math>e^{ix} = \cos x + i\sin x.</math>
Additionally,
:<math>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==
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]].
Relationships to ordinary trigonometric functions are given by [[Euler's formula]] for complex numbers:
:<math>\begin{align}
e^{i x} &= \cos x + i \sin x \\
e^{-i x} &= \cos x - i \sin x
\end{align}</math>
so:
:<math>\begin{align}
\cosh(ix) &= \frac{1}{2} \left(e^{i x} + e^{-i x}\right) = \cos x \\
\sinh(ix) &= \frac{1}{2} \left(e^{i x} - e^{-i x}\right) = i \sin x \\
\cosh(x+iy) &= \cosh(x) \cos(y) + i \sinh(x) \sin(y) \\
\sinh(x+iy) &= \sinh(x) \cos(y) + i \cosh(x) \sin(y) \\
\tanh(ix) &= i \tan x \\
\cosh x &= \cos(ix) \\
\sinh x &= - i \sin(ix) \\
\tanh x &= - i \tan(ix)
\end{align}</math>
Thus, hyperbolic functions are [[periodic function|periodic]] with respect to the imaginary component, with period <math>2 \pi i</math> (<math>\pi i</math> for hyperbolic tangent and cotangent).
{| style="text-align:center"
|+ Hyperbolic functions in the complex plane
|[[Image:Complex Sinh.jpg|1000x140px|none]]
|[[Image:Complex Cosh.jpg|1000x140px|none]]
|[[Image:Complex Tanh.jpg|1000x140px|none]]
|[[Image:Complex Coth.jpg|1000x140px|none]]
|[[Image:Complex Sech.jpg|1000x140px|none]]
|[[Image:Complex Csch.jpg|1000x140px|none]]
|-
|<math>\operatorname{sinh}(z)</math>
|<math>\operatorname{cosh}(z)</math>
|<math>\operatorname{tanh}(z)</math>
|<math>\operatorname{coth}(z)</math>
|<math>\operatorname{sech}(z)</math>
|<math>\operatorname{csch}(z)</math>
|}-->
== Referensi ==
<references/>{{Daftar fungsi matematika}}{{Authority control}}
[[Kategori:Matematika]]
| |||