Hyperbolic functions

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.{{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}}

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.{{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}}

In 1772, Roger Cotes suggested to modify the trigonometric functions using the imaginary unit $i=\backslash sqrt\{-1\}$ to obtain an oblate spheroid from a prolate one.

Hyperbolic functions were formally introduced in the 1757 by Vincenzo Riccati. 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. 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.

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

{{main|Trigonometric functions#Notation}}

File:sinh cosh tanh.svg

File:csch sech coth.svg

There are various equivalent ways to define the hyperbolic functions.

File:Hyperbolic and exponential; sinh.svg of {{math|e^x}} and {{math|e^−x}}]]

File:Hyperbolic and exponential; cosh.svg of {{math|e^x}} and {{math|e^−x}}]]

In terms of the exponential function:

• Hyperbolic sine: the odd part of the exponential function, that is, $$\backslash sinh\; x\; =\; \backslash frac\; \{e^x\; -\; e^\{-x\}\}\; \{2\}\; =\; \backslash frac\; \{e^\{2x\}\; -\; 1\}\; \{2e^x\}\; =\; \backslash frac\; \{1\; -\; e^\{-2x\}\}\; \{2e^\{-x\}\}.$$
• Hyperbolic cosine: the even part of the exponential function, that is, $$\backslash cosh\; x\; =\; \backslash frac\; \{e^x\; +\; e^\{-x\}\}\; \{2\}\; =\; \backslash frac\; \{e^\{2x\}\; +\; 1\}\; \{2e^x\}\; =\; \backslash frac\; \{1\; +\; e^\{-2x\}\}\; \{2e^\{-x\}\}.$$
• Hyperbolic tangent: $$\backslash tanh\; x\; =\; \backslash frac\{\backslash sinh\; x\}\{\backslash cosh\; x\}\; =\; \backslash frac\; \{e^x\; -\; e^\{-x\}\}\; \{e^x\; +\; e^\{-x\}\}$$

= \frac{e^{2x} - 1} {e^{2x} + 1}.

• Hyperbolic cotangent: for {{math|x ≠ 0}}, $$\backslash coth\; x\; =\; \backslash frac\{\backslash cosh\; x\}\{\backslash sinh\; x\}\; =\; \backslash frac\; \{e^x\; +\; e^\{-x\}\}\; \{e^x\; -\; e^\{-x\}\}$$

= \frac{e^{2x} + 1} {e^{2x} - 1}.

• Hyperbolic secant: $$\backslash operatorname\{sech\}\; x\; =\; \backslash frac\{1\}\{\backslash cosh\; x\}\; =\; \backslash frac\; \{2\}\; \{e^x\; +\; e^\{-x\}\}$$

= \frac{2e^x} {e^{2x} + 1}.

• Hyperbolic cosecant: for {{math|x ≠ 0}}, $$\backslash operatorname\{csch\}\; x\; =\; \backslash frac\{1\}\{\backslash sinh\; x\}\; =\; \backslash frac\; \{2\}\; \{e^x\; -\; e^\{-x\}\}$$

= \frac{2e^x} {e^{2x} - 1}.

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

$$\backslash begin\{align\}$$

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

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

\end{align}

with the initial conditions $s(0)\; =\; 0,\; c(0)\; =\; 1.$ The initial conditions make the solution unique; without them any pair of functions $(a\; e^x\; +\; b\; e^\{-x\},\; a\; e^x\; -\; b\; e^\{-x\})$ 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.

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

• Hyperbolic sine: $$\backslash sinh\; x\; =\; -i\; \backslash sin\; (i\; x).$$
• Hyperbolic cosine: $$\backslash cosh\; x\; =\; \backslash cos\; (i\; x).$$
• Hyperbolic tangent: $$\backslash tanh\; x\; =\; -i\; \backslash tan\; (i\; x).$$
• Hyperbolic cotangent: $$\backslash coth\; x\; =\; i\; \backslash cot\; (i\; x).$$
• Hyperbolic secant: $$\backslash operatorname\{sech\}\; x\; =\; \backslash sec\; (i\; x).$$
• Hyperbolic cosecant:$$\backslash operatorname\{csch\}\; x\; =\; i\; \backslash csc\; (i\; x).$$

where {{mvar|i}} is the imaginary unit with {{math|1=i² = −1}}.

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

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:{{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}}

$$\backslash text\{area\}\; =\; \backslash int\_a^b\; \backslash cosh\; x\; \backslash ,dx\; =\; \backslash int\_a^b\; \backslash sqrt\{1\; +\; \backslash left(\backslash frac\{d\}\{dx\}\; \backslash cosh\; x\; \backslash right)^2\}\; \backslash ,dx\; =\; \backslash text\{arc\; length.\}$$

The hyperbolic tangent is the (unique) solution to the differential equation {{math|1=f ′ = 1 − f ²}}, with {{math|1=f (0) = 0}}.{{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)]{{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]

{{Anchor|Osborn}}

The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborn's rule states that one can convert any trigonometric identity (up to but not including sinhs or implied sinhs of 4th degree) for $\backslash theta$, $2\backslash theta$, $3\backslash theta$ or $\backslash theta$ and $\backslash varphi$ 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:

$$\backslash begin\{align\}$$

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

\cosh (-x) &= \cosh x

\end{align}

Hence:

$$\backslash begin\{align\}$$

\tanh (-x) &= -\tanh x \\

\coth (-x) &= -\coth x \\

\operatorname{sech} (-x) &= \operatorname{sech} x \\

\operatorname{csch} (-x) &= -\operatorname{csch} x

\end{align}

Thus, {{math|cosh x}} and {{math|sech x}} are even functions; the others are odd functions.

$$\backslash begin\{align\}$$

\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}

Hyperbolic sine and cosine satisfy:

$$\backslash begin\{align\}$$

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

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

\end{align}

which are analogous to Euler's formula, and

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

which is analogous to the Pythagorean trigonometric identity.

One also has

$$\backslash begin\{align\}$$

\operatorname{sech} ^{2} x &= 1 - \tanh^{2} x \\

\operatorname{csch} ^{2} x &= \coth^{2} x - 1

\end{align}

for the other functions.

$$\backslash begin\{align\}$$

\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}

particularly

$$\backslash begin\{align\}$$

\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}

Also:

$$\backslash begin\{align\}$$

\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}

$$\backslash begin\{align\}$$

\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}

Also:{{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.}}

$$\backslash begin\{align\}$$

\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}

$$\backslash begin\{align\}$$

\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}

where {{math|sgn}} is the sign function.

If {{math|x ≠ 0}}, then{{cite web|title=Prove the identity tanh(x/2) = (cosh(x) - 1)/sinh(x) | url=https://math.stackexchange.com/q/1565753 |website=StackExchange (mathematics) | access-date=24 January 2016}}

$$\backslash tanh\backslash left(\backslash frac\{x\}\{2\}\backslash right)\; =\; \backslash frac\{\backslash cosh\; x\; -\; 1\}\{\backslash sinh\; x\}\; =\; \backslash coth\; x\; -\; \backslash operatorname\{csch\}\; x$$

$$\backslash begin\{align\}$$

\sinh^2 x &= \tfrac{1}{2}(\cosh 2x - 1) \\

\cosh^2 x &= \tfrac{1}{2}(\cosh 2x + 1)

\end{align}

The following inequality is useful in statistics:{{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]

$$\backslash operatorname\{cosh\}(t)\; \backslash leq\; e^\{t^2\; /2\}.$$

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

{{main|Inverse hyperbolic function}}

$$\backslash begin\{align\}$$

\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}

$$\backslash begin\{align\}$$

\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}

$$\backslash begin\{align\}$$

\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}

Each of the functions {{math|sinh}} and {{math|cosh}} is equal to its second derivative, that is:

$$\backslash frac\{d^2\}\{dx^2\}\backslash sinh\; x\; =\; \backslash sinh\; x$$

$$\backslash frac\{d^2\}\{dx^2\}\backslash cosh\; x\; =\; \backslash cosh\; x\; \backslash ,\; .$$

All functions with this property are linear combinations of {{math|sinh}} and {{math|cosh}}, in particular the exponential functions $e^x$ and $e^\{-x\}$.{{dlmf|id=4.34}}

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

$$\backslash begin\{align\}$$

\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}

The following integrals can be proved using hyperbolic substitution:

$$\backslash begin\{align\}$$

\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}

where C is the constant of integration.

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.

$$\backslash sinh\; x\; =\; x\; +\; \backslash frac\; \{x^3\}\; \{3!\}\; +\; \backslash frac\; \{x^5\}\; \{5!\}\; +\; \backslash frac\; \{x^7\}\; \{7!\}\; +\; \backslash cdots\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{x^\{2n+1\}\}\{(2n+1)!\}$$

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.

$$\backslash cosh\; x\; =\; 1\; +\; \backslash frac\; \{x^2\}\; \{2!\}\; +\; \backslash frac\; \{x^4\}\; \{4!\}\; +\; \backslash frac\; \{x^6\}\; \{6!\}\; +\; \backslash cdots\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{x^\{2n\}\}\{(2n)!\}$$

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.

$$\backslash begin\{align\}$$

\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}

where:

• $B\_n$ is the nth Bernoulli number
• $E\_n$ is the nth Euler number

The following expansions are valid in the whole complex plane:

:$\backslash sinh\; x\; =\; x\backslash prod\_\{n=1\}^\backslash infty\backslash left(1+\backslash frac\{x^2\}\{n^2\backslash pi^2\}\backslash right)\; =$

\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}}}}

:$\backslash cosh\; x\; =\; \backslash prod\_\{n=1\}^\backslash infty\backslash left(1+\backslash frac\{x^2\}\{(n-1/2)^2\backslash pi^2\}\backslash right)\; =\; \backslash cfrac\{1\}\{1\; -\; \backslash cfrac\{x^2\}\{1\; \backslash cdot\; 2\; +\; x^2\; -\; \backslash cfrac\{1\; \backslash cdot\; 2x^2\}\{3\; \backslash cdot\; 4\; +\; x^2\; -\; \backslash cfrac\{3\; \backslash cdot\; 4x^2\}\{5\; \backslash cdot\; 6\; +\; x^2\; -\; \backslash ddots\}\}\}\}$

:$\backslash tanh\; x\; =\; \backslash cfrac\{1\}\{\backslash cfrac\{1\}\{x\}\; +\; \backslash cfrac\{1\}\{\backslash cfrac\{3\}\{x\}\; +\; \backslash cfrac\{1\}\{\backslash cfrac\{5\}\{x\}\; +\; \backslash cfrac\{1\}\{\backslash cfrac\{7\}\{x\}\; +\; \backslash ddots\}\}\}\}$

File:Circular and hyperbolic angle.svg 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=r²u/2}}, it will be equal to {{mvar|u}} when {{math|1=r = {{sqrt|2}}}}. In the diagram, such a circle is tangent to the hyperbola xy = 1 at (1,1). The yellow sector depicts an area and angle magnitude. Similarly, the yellow and red regions together depict a hyperbolic sector with area corresponding to hyperbolic angle magnitude.

The legs of the two right triangles 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.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]

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 {{math|a cosh(x/a)}} is the catenary, the curve formed by a uniform flexible chain, hanging freely between two fixed points under uniform gravity.

The decomposition of the exponential function in its even and odd parts gives the identities

$$e^x\; =\; \backslash cosh\; x\; +\; \backslash sinh\; x,$$

and

$$e^\{-x\}\; =\; \backslash cosh\; x\; -\; \backslash sinh\; x.$$

Combined with Euler's formula

$$e^\{ix\}\; =\; \backslash cos\; x\; +\; i\backslash sin\; x,$$

this gives

$$e^\{x+iy\}=(\backslash cosh\; x+\backslash sinh\; x)(\backslash cos\; y+i\backslash sin\; y)$$

for the general complex exponential function.

Additionally,

$$e^x\; =\; \backslash sqrt\{\backslash frac\{1\; +\; \backslash tanh\; x\}\{1\; -\; \backslash tanh\; x\}\}\; =\; \backslash frac\{1\; +\; \backslash tanh\; \backslash frac\{x\}\{2\}\}\{1\; -\; \backslash tanh\; \backslash frac\{x\}\{2\}\}$$

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:

$$\backslash begin\{align\}$$

e^{i x} &= \cos x + i \sin x \\

e^{-i x} &= \cos x - i \sin x

\end{align}

so:

$$\backslash begin\{align\}$$

\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}

Thus, hyperbolic functions are periodic with respect to the imaginary component, with period $2\; \backslash pi\; i$ ($\backslash pi\; i$ for hyperbolic tangent and cotangent).

• e (mathematical constant)
• Equal incircles theorem, based on sinh
• Hyperbolastic functions
• Hyperbolic growth
• Inverse hyperbolic functions
• List of integrals of hyperbolic functions
• Poinsot's spirals
• Sigmoid function
• Soboleva modified hyperbolic tangent
• Trigonometric functions

{{Reflist|refs=

{{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 }}

}}

{{Commons category|Hyperbolic functions}}

• {{springer|title=Hyperbolic functions|id=p/h048250}}
• [http://planetmath.org/hyperbolicfunctions Hyperbolic functions] on PlanetMath
• [https://web.archive.org/web/20071006172054/http://glab.trixon.se/ GonioLab]: Visualization of the unit circle, trigonometric and hyperbolic functions (Java Web Start)
• [http://www.calctool.org/CALC/math/trigonometry/hyperbolic Web-based calculator of hyperbolic functions]

{{Trigonometric and hyperbolic functions}}

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{{DEFAULTSORT:Hyperbolic Function}}

Category:Exponentials

Category:Hyperbolic geometry

Category:Analytic functions