Hyperbolic functions

"Hyperbolic curve" redirects here. For the geometric curve, seeHyperbola.

Inmathematics,hyperbolic functionsare analogues of the ordinarytrigonometric functions,but defined using thehyperbolarather than thecircle.Just as the points(cost,sint)form acircle with a unit radius,the points(cosht,sinht)form the right half of theunit hyperbola.Also, similarly to how the derivatives ofsin(t)andcos(t)arecos(t)and–sin(t)respectively, the derivatives ofsinh(t)andcosh(t)arecosh(t)and+sinh(t)respectively.

Hyperbolic functions occur in the calculations of angles and distances inhyperbolic geometry.They also occur in the solutions of many lineardifferential equations(such as the equation defining acatenary),cubic equations,andLaplace's equationinCartesian coordinates.Laplace's equationsare important in many areas ofphysics,includingelectromagnetic theory,heat transfer,fluid dynamics,andspecial relativity.

The basic hyperbolic functions are:

• hyperbolic sine"sinh"(/ˈsɪŋ,ˈsɪntʃ,ˈʃaɪn/),
• hyperbolic cosine"cosh"(/ˈkɒʃ,ˈkoʊʃ/),

from which are derived:

• hyperbolic tangent"tanh"(/ˈtæŋ,ˈtæntʃ,ˈθæn/),
• hyperbolic cotangent"coth"(/ˈkɒθ,ˈkoʊθ/),
• hyperbolic secant"sech"(/ˈsɛtʃ,ˈʃɛk/),
• hyperbolic cosecant"csch"or"cosech"(/ˈkoʊsɛtʃ,ˈkoʊʃɛk/)

corresponding to the derived trigonometric functions.

Theinverse hyperbolic functionsare:

• area hyperbolic sine"arsinh"(also denoted"sinh⁻¹","asinh"or sometimes"arcsinh")
• area hyperbolic cosine"arcosh"(also denoted"cosh⁻¹","acosh"or sometimes"arccosh")
• area hyperbolic tangent"artanh"(also denoted"tanh⁻¹","atanh"or sometimes"arctanh")
• area hyperbolic cotangent"arcoth"(also denoted"coth⁻¹","acoth"or sometimes"arccoth")
• area hyperbolic secant"arsech"(also denoted"sech⁻¹","asech"or sometimes"arcsech")
• area hyperbolic cosecant"arcsch"(also denoted"arcosech","csch⁻¹","cosech⁻¹","acsch","acosech",or sometimes"arccsch"or"arccosech")

Araythrough theunit hyperbolax²−y²= 1at the point(cosha,sinha),whereais twice the area between the ray, the hyperbola, and thex-axis. For points on the hyperbola below thex-axis, the area is considered negative (seeanimated versionwith comparison with the trigonometric (circular) functions).

The hyperbolic functions take areal argumentcalled ahyperbolic angle.The size of a hyperbolic angle is twice the area of itshyperbolic sector.The hyperbolic functions may be defined in terms of thelegs of a right trianglecovering this sector.

Incomplex analysis,the hyperbolic functions arise when applying the ordinary sine and cosine functions to an imaginary angle. The hyperbolic sine and the hyperbolic cosine areentire functions.As a result, the other hyperbolic functions aremeromorphicin the whole complex plane.

ByLindemann–Weierstrass theorem,the hyperbolic functions have atranscendental valuefor every non-zeroalgebraic valueof the argument.

Hyperbolic functions were introduced in the 1760s independently byVincenzo RiccatiandJohann Heinrich Lambert.Riccati usedSc.andCc.(sinus/cosinus circulare) to refer to circular functions andSh.andCh.(sinus/cosinus hyperbolico) to refer to hyperbolic functions. Lambert adopted the names, but altered the abbreviations to those used today.The abbreviationssh,ch,th,cthare also currently used, depending on personal preference.

Notation

Main article:Trigonometric functions § Notation

Definitions

sinh,coshandtanh

csch,sechandcoth

There are various equivalent ways to define the hyperbolic functions.

Exponential definitions

sinhxis half thedifferenceofe^xande^−x

coshxis theaverageofe^xande^−x

In terms of theexponential function:

• Hyperbolic sine: theodd partof the exponential function, that is,$$\sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{2x}-1}{2e^{x}}}={\frac {1-e^{-2x}}{2e^{-x}}}.$$
• Hyperbolic cosine: theeven partof the exponential function, that is,$$\cosh x={\frac {e^{x}+e^{-x}}{2}}={\frac {e^{2x}+1}{2e^{x}}}={\frac {1+e^{-2x}}{2e^{-x}}}.$$
• Hyperbolic tangent:$$\tanh x={\frac {\sinh x}{\cosh x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}={\frac {e^{2x}-1}{e^{2x}+1}}.$$
• Hyperbolic cotangent: forx≠ 0,$$\coth x={\frac {\cosh x}{\sinh x}}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}={\frac {e^{2x}+1}{e^{2x}-1}}.$$
• Hyperbolic secant:$$\operatorname {sech} x={\frac {1}{\cosh x}}={\frac {2}{e^{x}+e^{-x}}}={\frac {2e^{x}}{e^{2x}+1}}.$$
• Hyperbolic cosecant: forx≠ 0,$$\operatorname {csch} x={\frac {1}{\sinh x}}={\frac {2}{e^{x}-e^{-x}}}={\frac {2e^{x}}{e^{2x}-1}}.$$

Differential equation definitions

The hyperbolic functions may be defined as solutions ofdifferential equations:The hyperbolic sine and cosine are the solution(s,c)of the system $${\begin{aligned}c'(x)&=s(x),\\s'(x)&=c(x),\\\end{aligned}}$$ with the initial conditions$s(0)=0,c(0)=1.$The initial conditions make the solution unique; without them any pair of functions$(ae^{x}+be^{-x},ae^{x}-be^{-x})$would be a solution.

sinh(x)andcosh(x)are also the unique solution of the equationf ″(x) =f (x), such thatf (0) = 1,f ′(0) = 0for the hyperbolic cosine, andf (0) = 0,f ′(0) = 1for the hyperbolic sine.

Complex trigonometric definitions

Hyperbolic functions may also be deduced fromtrigonometric functionswithcomplexarguments:

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

whereiis theimaginary unitwithi²= −1.

The above definitions are related to the exponential definitions viaEuler's formula(See§ Hyperbolic functions for complex numbersbelow).

Characterizing properties

Hyperbolic cosine

It can be shown that thearea under the curveof the hyperbolic cosine (over a finite interval) is always equal to thearc lengthcorresponding to that interval: $${\text{area}}=\int _{a}^{b}\cosh x\,dx=\int _{a}^{b}{\sqrt {1+\left({\frac {d}{dx}}\cosh x\right)^{2}}}\,dx={\text{arc length.}}$$

Hyperbolic tangent

The hyperbolic tangent is the (unique) solution to thedifferential equationf ′ = 1 −f ²,withf (0) = 0.

Useful relations

The hyperbolic functions satisfy many identities, all of them similar in form to thetrigonometric identities.In fact,Osborn's rulestates that one can convert any trigonometric identity (up to but not including sinhs or implied sinhs of 4th degree) for$\theta$,$2\theta$,$3\theta$or$\theta$and$\varphi$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: $${\begin{aligned}\sinh(-x)&=-\sinh x\\\cosh(-x)&=\cosh x\end{aligned}}$$

Hence: $${\begin{aligned}\tanh(-x)&=-\tanh x\\\coth(-x)&=-\coth x\\\operatorname {sech} (-x)&=\operatorname {sech} x\\\operatorname {csch} (-x)&=-\operatorname {csch} x\end{aligned}}$$

Thus,coshxandsechxareeven functions;the others areodd functions.

$${\begin{aligned}\operatorname {arsech} x&=\operatorname {arcosh} \left({\frac {1}{x}}\right)\\\operatorname {arcsch} x&=\operatorname {arsinh} \left({\frac {1}{x}}\right)\\\operatorname {arcoth} x&=\operatorname {artanh} \left({\frac {1}{x}}\right)\end{aligned}}$$

Hyperbolic sine and cosine satisfy: $${\begin{aligned}\cosh x+\sinh x&=e^{x}\\\cosh x-\sinh x&=e^{-x}\\\cosh ^{2}x-\sinh ^{2}x&=1\end{aligned}}$$

the last of which is similar to thePythagorean trigonometric identity.

One also has $${\begin{aligned}\operatorname {sech} ^{2}x&=1-\tanh ^{2}x\\\operatorname {csch} ^{2}x&=\coth ^{2}x-1\end{aligned}}$$

for the other functions.

Sums of arguments

$${\begin{aligned}\sinh(x+y)&=\sinh x\cosh y+\cosh x\sinh y\\\cosh(x+y)&=\cosh x\cosh y+\sinh x\sinh y\\\tanh(x+y)&={\frac {\tanh x+\tanh y}{1+\tanh x\tanh y}}\\\end{aligned}}$$ particularly $${\begin{aligned}\cosh(2x)&=\sinh ^{2}{x}+\cosh ^{2}{x}=2\sinh ^{2}x+1=2\cosh ^{2}x-1\\\sinh(2x)&=2\sinh x\cosh x\\\tanh(2x)&={\frac {2\tanh x}{1+\tanh ^{2}x}}\\\end{aligned}}$$

Also: $${\begin{aligned}\sinh x+\sinh y&=2\sinh \left({\frac {x+y}{2}}\right)\cosh \left({\frac {x-y}{2}}\right)\\\cosh x+\cosh y&=2\cosh \left({\frac {x+y}{2}}\right)\cosh \left({\frac {x-y}{2}}\right)\\\end{aligned}}$$

Subtraction formulas

$${\begin{aligned}\sinh(x-y)&=\sinh x\cosh y-\cosh x\sinh y\\\cosh(x-y)&=\cosh x\cosh y-\sinh x\sinh y\\\tanh(x-y)&={\frac {\tanh x-\tanh y}{1-\tanh x\tanh y}}\\\end{aligned}}$$

Also: $${\begin{aligned}\sinh x-\sinh y&=2\cosh \left({\frac {x+y}{2}}\right)\sinh \left({\frac {x-y}{2}}\right)\\\cosh x-\cosh y&=2\sinh \left({\frac {x+y}{2}}\right)\sinh \left({\frac {x-y}{2}}\right)\\\end{aligned}}$$

Half argument formulas

$${\begin{aligned}\sinh \left({\frac {x}{2}}\right)&={\frac {\sinh x}{\sqrt {2(\cosh x+1)}}}&&=\operatorname {sgn} x\,{\sqrt {\frac {\cosh x-1}{2}}}\\[6px]\cosh \left({\frac {x}{2}}\right)&={\sqrt {\frac {\cosh x+1}{2}}}\\[6px]\tanh \left({\frac {x}{2}}\right)&={\frac {\sinh x}{\cosh x+1}}&&=\operatorname {sgn} x\,{\sqrt {\frac {\cosh x-1}{\cosh x+1}}}={\frac {e^{x}-1}{e^{x}+1}}\end{aligned}}$$

wheresgnis thesign function.

Ifx≠ 0,then

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

Square formulas

$${\begin{aligned}\sinh ^{2}x&={\tfrac {1}{2}}(\cosh 2x-1)\\\cosh ^{2}x&={\tfrac {1}{2}}(\cosh 2x+1)\end{aligned}}$$

Inequalities

The following inequality is useful in statistics: $$\operatorname {cosh} (t)\leq e^{t^{2}/2}.$$

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

Inverse functions as logarithms

Main article:Inverse hyperbolic function

$${\begin{aligned}\operatorname {arsinh} (x)&=\ln \left(x+{\sqrt {x^{2}+1}}\right)\\\operatorname {arcosh} (x)&=\ln \left(x+{\sqrt {x^{2}-1}}\right)&&x\geq 1\\\operatorname {artanh} (x)&={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right)&&|x|<1\\\operatorname {arcoth} (x)&={\frac {1}{2}}\ln \left({\frac {x+1}{x-1}}\right)&&|x|>1\\\operatorname {arsech} (x)&=\ln \left({\frac {1}{x}}+{\sqrt {{\frac {1}{x^{2}}}-1}}\right)=\ln \left({\frac {1+{\sqrt {1-x^{2}}}}{x}}\right)&&0<x\leq 1\\\operatorname {arcsch} (x)&=\ln \left({\frac {1}{x}}+{\sqrt {{\frac {1}{x^{2}}}+1}}\right)&&x\neq 0\end{aligned}}$$

Derivatives

$${\begin{aligned}{\frac {d}{dx}}\sinh x&=\cosh x\\{\frac {d}{dx}}\cosh x&=\sinh x\\{\frac {d}{dx}}\tanh x&=1-\tanh ^{2}x=\operatorname {sech} ^{2}x={\frac {1}{\cosh ^{2}x}}\\{\frac {d}{dx}}\coth x&=1-\coth ^{2}x=-\operatorname {csch} ^{2}x=-{\frac {1}{\sinh ^{2}x}}&&x\neq 0\\{\frac {d}{dx}}\operatorname {sech} x&=-\tanh x\operatorname {sech} x\\{\frac {d}{dx}}\operatorname {csch} x&=-\coth x\operatorname {csch} x&&x\neq 0\end{aligned}}$$ $${\begin{aligned}{\frac {d}{dx}}\operatorname {arsinh} x&={\frac {1}{\sqrt {x^{2}+1}}}\\{\frac {d}{dx}}\operatorname {arcosh} x&={\frac {1}{\sqrt {x^{2}-1}}}&&1<x\\{\frac {d}{dx}}\operatorname {artanh} x&={\frac {1}{1-x^{2}}}&&|x|<1\\{\frac {d}{dx}}\operatorname {arcoth} x&={\frac {1}{1-x^{2}}}&&1<|x|\\{\frac {d}{dx}}\operatorname {arsech} x&=-{\frac {1}{x{\sqrt {1-x^{2}}}}}&&0<x<1\\{\frac {d}{dx}}\operatorname {arcsch} x&=-{\frac {1}{|x|{\sqrt {1+x^{2}}}}}&&x\neq 0\end{aligned}}$$

Second derivatives

Each of the functionssinhandcoshis equal to itssecond derivative,that is: $${\frac {d^{2}}{dx^{2}}}\sinh x=\sinh x$$ $${\frac {d^{2}}{dx^{2}}}\cosh x=\cosh x\,.$$

All functions with this property arelinear combinationsofsinhandcosh,in particular theexponential functions$e^{x}$and$e^{-x}$.

Standard integrals

For a full list, seelist of integrals of hyperbolic functions.

$${\begin{aligned}\int \sinh(ax)\,dx&=a^{-1}\cosh(ax)+C\\\int \cosh(ax)\,dx&=a^{-1}\sinh(ax)+C\\\int \tanh(ax)\,dx&=a^{-1}\ln(\cosh(ax))+C\\\int \coth(ax)\,dx&=a^{-1}\ln \left|\sinh(ax)\right|+C\\\int \operatorname {sech} (ax)\,dx&=a^{-1}\arctan(\sinh(ax))+C\\\int \operatorname {csch} (ax)\,dx&=a^{-1}\ln \left|\tanh \left({\frac {ax}{2}}\right)\right|+C=a^{-1}\ln \left|\coth \left(ax\right)-\operatorname {csch} \left(ax\right)\right|+C=-a^{-1}\operatorname {arcoth} \left(\cosh \left(ax\right)\right)+C\end{aligned}}$$

The following integrals can be proved usinghyperbolic substitution: $${\begin{aligned}\int {{\frac {1}{\sqrt {a^{2}+u^{2}}}}\,du}&=\operatorname {arsinh} \left({\frac {u}{a}}\right)+C\\\int {{\frac {1}{\sqrt {u^{2}-a^{2}}}}\,du}&=\operatorname {sgn} {u}\operatorname {arcosh} \left|{\frac {u}{a}}\right|+C\\\int {\frac {1}{a^{2}-u^{2}}}\,du&=a^{-1}\operatorname {artanh} \left({\frac {u}{a}}\right)+C&&u^{2}<a^{2}\\\int {\frac {1}{a^{2}-u^{2}}}\,du&=a^{-1}\operatorname {arcoth} \left({\frac {u}{a}}\right)+C&&u^{2}>a^{2}\\\int {{\frac {1}{u{\sqrt {a^{2}-u^{2}}}}}\,du}&=-a^{-1}\operatorname {arsech} \left|{\frac {u}{a}}\right|+C\\\int {{\frac {1}{u{\sqrt {a^{2}+u^{2}}}}}\,du}&=-a^{-1}\operatorname {arcsch} \left|{\frac {u}{a}}\right|+C\end{aligned}}$$

whereCis theconstant of integration.

Taylor series expressions

It is possible to express explicitly theTaylor seriesat zero (or theLaurent series,if the function is not defined at zero) of the above functions.

$$\sinh x=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}$$ This series isconvergentfor everycomplexvalue ofx.Since the functionsinhxisodd,only odd exponents forxoccur in its Taylor series.

$$\cosh x=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}$$ This series isconvergentfor everycomplexvalue ofx.Since the functioncoshxiseven,only even exponents forxoccur in its Taylor series.

The sum of the sinh and cosh series is theinfinite seriesexpression of theexponential function.

The following series are followed by a description of a subset of theirdomain of convergence,where the series is convergent and its sum equals the function. $${\begin{aligned}\tanh x&=x-{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}-{\frac {17x^{7}}{315}}+\cdots =\sum _{n=1}^{\infty }{\frac {2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}},\qquad \left|x\right|<{\frac {\pi }{2}}\\\coth x&=x^{-1}+{\frac {x}{3}}-{\frac {x^{3}}{45}}+{\frac {2x^{5}}{945}}+\cdots =\sum _{n=0}^{\infty }{\frac {2^{2n}B_{2n}x^{2n-1}}{(2n)!}},\qquad 0<\left|x\right|<\pi \\\operatorname {sech} x&=1-{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}-{\frac {61x^{6}}{720}}+\cdots =\sum _{n=0}^{\infty }{\frac {E_{2n}x^{2n}}{(2n)!}},\qquad \left|x\right|<{\frac {\pi }{2}}\\\operatorname {csch} x&=x^{-1}-{\frac {x}{6}}+{\frac {7x^{3}}{360}}-{\frac {31x^{5}}{15120}}+\cdots =\sum _{n=0}^{\infty }{\frac {2(1-2^{2n-1})B_{2n}x^{2n-1}}{(2n)!}},\qquad 0<\left|x\right|<\pi \end{aligned}}$$

where:

• $B_{n}$is thenthBernoulli number
• $E_{n}$is thenthEuler number

Infinite products and continued fractions

The following expansions are valid in the whole complex plane:

$\sinh x=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{n^{2}\pi ^{2}}}\right)={\cfrac {x}{1-{\cfrac {x^{2}}{2\cdot 3+x^{2}-{\cfrac {2\cdot 3x^{2}}{4\cdot 5+x^{2}-{\cfrac {4\cdot 5x^{2}}{6\cdot 7+x^{2}-\ddots }}}}}}}}$

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

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

Comparison with circular functions

Circle and hyperbola tangent at (1,1) display geometry of circular functions in terms ofcircular sectorareauand hyperbolic functions depending onhyperbolic sectorareau.

The hyperbolic functions represent an expansion oftrigonometrybeyond thecircular functions.Both types depend on anargument,eithercircular angleorhyperbolic angle.

Since thearea of a circular sectorwith radiusrand angleu(in radians) isr²u/2,it will be equal touwhenr=√2.In the diagram, such a circle is tangent to the hyperbolaxy= 1 at (1,1). The yellow sector depicts an area and angle magnitude. Similarly, the yellow and red regions together depict ahyperbolic sectorwith area corresponding to hyperbolic angle magnitude.

The legs of the tworight triangleswith hypotenuse on the ray defining the angles are of length√2times the circular and hyperbolic functions.

The hyperbolic angle is aninvariant measurewith respect to thesqueeze mapping,just as the circular angle is invariant under rotation.

TheGudermannian functiongives a direct relationship between the circular functions and the hyperbolic functions that does not involve complex numbers.

The graph of the functionacosh(x/a)is thecatenary,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 itseven and odd partsgives the identities $$e^{x}=\cosh x+\sinh x,$$ and $$e^{-x}=\cosh x-\sinh x.$$ Combined withEuler's formula $$e^{ix}=\cos x+i\sin x,$$ this gives $$e^{x+iy}=(\cosh x+\sinh x)(\cos y+i\sin y)$$ for thegeneral complex exponential function.

Additionally, $$e^{x}={\sqrt {\frac {1+\tanh x}{1-\tanh x}}}={\frac {1+\tanh {\frac {x}{2}}}{1-\tanh {\frac {x}{2}}}}$$

Hyperbolic functions for complex numbers

Hyperbolic functions in the complex plane

$\sinh(z)$	$\cosh(z)$	$\tanh(z)$	$\coth(z)$	$\operatorname {sech} (z)$	$\operatorname {csch} (z)$

Since theexponential functioncan be defined for anycomplexargument, we can also extend the definitions of the hyperbolic functions to complex arguments. The functionssinhzandcoshzare thenholomorphic.

Relationships to ordinary trigonometric functions are given byEuler's formulafor complex numbers: $${\begin{aligned}e^{ix}&=\cos x+i\sin x\\e^{-ix}&=\cos x-i\sin x\end{aligned}}$$ so: $${\begin{aligned}\cosh(ix)&={\frac {1}{2}}\left(e^{ix}+e^{-ix}\right)=\cos x\\\sinh(ix)&={\frac {1}{2}}\left(e^{ix}-e^{-ix}\right)=i\sin x\\\cosh(x+iy)&=\cosh(x)\cos(y)+i\sinh(x)\sin(y)\\\sinh(x+iy)&=\sinh(x)\cos(y)+i\cosh(x)\sin(y)\\\tanh(ix)&=i\tan x\\\cosh x&=\cos(ix)\\\sinh x&=-i\sin(ix)\\\tanh x&=-i\tan(ix)\end{aligned}}$$

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

See also

• 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

External links

• "Hyperbolic functions",Encyclopedia of Mathematics,EMS Press,2001 [1994]
• Hyperbolic functionsonPlanetMath
• GonioLab:Visualization of the unit circle, trigonometric and hyperbolic functions (Java Web Start)
• Web-based calculator of hyperbolic functions