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Hyperbolic functions

From Wikipedia, the free encyclopedia
Collective name of 6 mathematical functions
"Hyperbolic curve" redirects here. For the geometric curve, seeHyperbola.

Inmathematics,hyperbolic functions are analogues of the ordinarytrigonometric functions, but defined using thehyperbola rather 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) andsinh(t) respectively.

Hyperbolic functions are used to express theangle of parallelism inhyperbolic geometry. They are used to expressLorentz boosts ashyperbolic rotations inspecial relativity. They also occur in the solutions of many lineardifferential equations (such as the equation defining acatenary),cubic equations, andLaplace's equation inCartesian coordinates.Laplace's equations are important in many areas ofphysics, includingelectromagnetic theory,heat transfer, andfluid dynamics.

The basic hyperbolic functions are:[1]

from which are derived:[4]

corresponding to the derived trigonometric functions.

Theinverse hyperbolic functions are:

  • inverse hyperbolic sine "arsinh" (also denoted "sinh−1", "asinh" or sometimes "arcsinh")[9][10][11]
  • inverse hyperbolic cosine "arcosh" (also denoted "cosh−1", "acosh" or sometimes "arccosh")
  • inverse hyperbolic tangent "artanh" (also denoted "tanh−1", "atanh" or sometimes "arctanh")
  • inverse hyperbolic cotangent "arcoth" (also denoted "coth−1", "acoth" or sometimes "arccoth")
  • inverse hyperbolic secant "arsech" (also denoted "sech−1", "asech" or sometimes "arcsech")
  • inverse hyperbolic cosecant "arcsch" (also denoted "arcosech", "csch−1", "cosech−1","acsch", "acosech", or sometimes "arccsch" or "arccosech")
Aray through theunit hyperbolax2 −y2 = 1 at the point(cosha, sinha), wherea is 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 version with comparison with the trigonometric (circular) functions).

The hyperbolic functions take arealargument called ahyperbolic angle. The magnitude of a hyperbolic angle is thearea of itshyperbolic sector toxy = 1. The hyperbolic functions may be defined in terms of thelegs of a right triangle covering 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 aremeromorphic in the whole complex plane.

ByLindemann–Weierstrass theorem, the hyperbolic functions have atranscendental value for every non-zeroalgebraic value of the argument.[12]

History

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The first known calculation of a hyperbolic trigonometry problem is attributed toGerardus Mercator when issuing theMercator map projection circa 1566. It requires tabulating solutions to a transcendental equation involving hyperbolic functions.[13]

The first to suggest a similarity between the sector of the circle and that of the hyperbola wasIsaac Newton in his 1687Principia Mathematica.[14]

Roger Cotes suggested to modify the trigonometric functions using theimaginary uniti=−1{\displaystyle i={\sqrt {-1}}} to obtain an oblatespheroid from a prolate one.[14]

Hyperbolic functions were formally introduced in 1757 byVincenzo Riccati.[14][13][15] Riccati usedSc. andCc. (sinus/cosinus circulare) to refer to circular functions andSh. andCh. (sinus/cosinus hyperbolico) to refer to hyperbolic functions.[14] As early as 1759,Daviet de Foncenex showed the interchangeability of the trigonometric and hyperbolic functions using the imaginary unit and extendedde Moivre's formula to hyperbolic functions.[15][14]

During the 1760s,Johann Heinrich Lambert systematized the use functions and provided exponential expressions in various publications.[14][15] Lambert credited Riccati for the terminology and names of the functions, but altered the abbreviations to those used today.[15][16]

Notation

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Main article:Trigonometric functions § Notation

Definitions

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sinh,cosh andtanh
csch,sech andcoth

There are various equivalent ways to define the hyperbolic functions.

Exponential definitions

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sinhx is half thedifference ofex ande−x
coshx is theaverage ofex ande−x

In terms of theexponential function:[1][4]

Differential equation definitions

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The hyperbolic functions may be defined as solutions ofdifferential equations: The hyperbolic sine and cosine are the solution(s,c) of the systemc′(x)=s(x),s′(x)=c(x),{\displaystyle {\begin{aligned}c'(x)&=s(x),\\s'(x)&=c(x),\\\end{aligned}}}with the initial conditionss(0)=0,c(0)=1.{\displaystyle s(0)=0,c(0)=1.} The initial conditions make the solution unique; without them any pair of functions(aex+be−x,aex−be−x){\displaystyle (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) = 0 for the hyperbolic cosine, andf (0) = 0,f ′(0) = 1 for the hyperbolic sine.

Complex trigonometric definitions

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Hyperbolic functions may also be deduced fromtrigonometric functions withcomplex arguments:

wherei is theimaginary unit withi2 = −1.

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

Characterizing properties

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Hyperbolic cosine

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It can be shown that thearea under the curve of the hyperbolic cosine (over a finite interval) is always equal to thearc length corresponding to that interval:[17]area=∫abcosh⁡xdx=∫ab1+(ddxcosh⁡x)2dx=arc length.{\displaystyle {\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

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The hyperbolic tangent is the (unique) solution to thedifferential equationf ′ = 1 −f 2, withf (0) = 0.[18][19]

Useful relations

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The hyperbolic functions satisfy many identities, all of them similar in form to thetrigonometric identities. In fact,Osborn's rule[20] states that one can convert any trigonometric identity (up to but not including sinhs or implied sinhs of 4th degree) forθ{\displaystyle \theta },2θ{\displaystyle 2\theta },3θ{\displaystyle 3\theta } orθ{\displaystyle \theta } andφ{\displaystyle \varphi } into a hyperbolic identity, by:

  1. expanding it completely in terms of integral powers of sines and cosines,
  2. changing sine to sinh and cosine to cosh, and
  3. switching the sign of every term containing a product of two sinhs.

Odd and even functions:sinh⁡(−x)=−sinh⁡xcosh⁡(−x)=cosh⁡x{\displaystyle {\begin{aligned}\sinh(-x)&=-\sinh x\\\cosh(-x)&=\cosh x\end{aligned}}}

Hence:tanh⁡(−x)=−tanh⁡xcoth⁡(−x)=−coth⁡xsech⁡(−x)=sech⁡xcsch⁡(−x)=−csch⁡x{\displaystyle {\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,coshx andsechx areeven functions; the others areodd functions.

arsech⁡x=arcosh⁡(1x)arcsch⁡x=arsinh⁡(1x)arcoth⁡x=artanh⁡(1x){\displaystyle {\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:cosh⁡x+sinh⁡x=excosh⁡x−sinh⁡x=e−x{\displaystyle {\begin{aligned}\cosh x+\sinh x&=e^{x}\\\cosh x-\sinh x&=e^{-x}\end{aligned}}}

which are analogous toEuler's formula, and

cosh2⁡x−sinh2⁡x=1{\displaystyle \cosh ^{2}x-\sinh ^{2}x=1}

which is analogous to thePythagorean trigonometric identity.

One also hassech2⁡x=1−tanh2⁡xcsch2⁡x=coth2⁡x−1{\displaystyle {\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

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sinh⁡(x+y)=sinh⁡xcosh⁡y+cosh⁡xsinh⁡ycosh⁡(x+y)=cosh⁡xcosh⁡y+sinh⁡xsinh⁡ytanh⁡(x+y)=tanh⁡x+tanh⁡y1+tanh⁡xtanh⁡y{\displaystyle {\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}}}particularlycosh⁡(2x)=sinh2⁡x+cosh2⁡x=2sinh2⁡x+1=2cosh2⁡x−1sinh⁡(2x)=2sinh⁡xcosh⁡xtanh⁡(2x)=2tanh⁡x1+tanh2⁡x{\displaystyle {\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:sinh⁡x+sinh⁡y=2sinh⁡(x+y2)cosh⁡(x−y2)cosh⁡x+cosh⁡y=2cosh⁡(x+y2)cosh⁡(x−y2){\displaystyle {\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

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sinh⁡(x−y)=sinh⁡xcosh⁡y−cosh⁡xsinh⁡ycosh⁡(x−y)=cosh⁡xcosh⁡y−sinh⁡xsinh⁡ytanh⁡(x−y)=tanh⁡x−tanh⁡y1−tanh⁡xtanh⁡y{\displaystyle {\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:[21]sinh⁡x−sinh⁡y=2cosh⁡(x+y2)sinh⁡(x−y2)cosh⁡x−cosh⁡y=2sinh⁡(x+y2)sinh⁡(x−y2){\displaystyle {\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

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sinh⁡(x2)=sinh⁡x2(cosh⁡x+1)=sgn⁡xcosh⁡x−12cosh⁡(x2)=cosh⁡x+12tanh⁡(x2)=sinh⁡xcosh⁡x+1=sgn⁡xcosh⁡x−1cosh⁡x+1=ex−1ex+1{\displaystyle {\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}}}

wheresgn is thesign function.

Ifx ≠ 0, then[22]

tanh⁡(x2)=cosh⁡x−1sinh⁡x=coth⁡x−csch⁡x{\displaystyle \tanh \left({\frac {x}{2}}\right)={\frac {\cosh x-1}{\sinh x}}=\coth x-\operatorname {csch} x}

Square formulas

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sinh2⁡x=12(cosh⁡2x−1)cosh2⁡x=12(cosh⁡2x+1){\displaystyle {\begin{aligned}\sinh ^{2}x&={\tfrac {1}{2}}(\cosh 2x-1)\\\cosh ^{2}x&={\tfrac {1}{2}}(\cosh 2x+1)\end{aligned}}}

Inequalities

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The following inequality is useful in statistics:[23]cosh⁡(t)≤et2/2.{\displaystyle \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

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Main article:Inverse hyperbolic function

arsinh⁡(x)=ln⁡(x+x2+1)arcosh⁡(x)=ln⁡(x+x2−1)x≥1artanh⁡(x)=12ln⁡(1+x1−x)|x|<1arcoth⁡(x)=12ln⁡(x+1x−1)|x|>1arsech⁡(x)=ln⁡(1x+1x2−1)=ln⁡(1+1−x2x)0<x≤1arcsch⁡(x)=ln⁡(1x+1x2+1)x≠0{\displaystyle {\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

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ddxsinh⁡x=cosh⁡xddxcosh⁡x=sinh⁡xddxtanh⁡x=1−tanh2⁡x=sech2⁡x=1cosh2⁡xddxcoth⁡x=1−coth2⁡x=−csch2⁡x=−1sinh2⁡xx≠0ddxsech⁡x=−tanh⁡xsech⁡xddxcsch⁡x=−coth⁡xcsch⁡xx≠0{\displaystyle {\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}}}ddxarsinh⁡x=1x2+1ddxarcosh⁡x=1x2−11<xddxartanh⁡x=11−x2|x|<1ddxarcoth⁡x=11−x21<|x|ddxarsech⁡x=−1x1−x20<x<1ddxarcsch⁡x=−1|x|1+x2x≠0{\displaystyle {\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

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Each of the functionssinh andcosh is equal to itssecond derivative, that is:d2dx2sinh⁡x=sinh⁡x{\displaystyle {\frac {d^{2}}{dx^{2}}}\sinh x=\sinh x}d2dx2cosh⁡x=cosh⁡x.{\displaystyle {\frac {d^{2}}{dx^{2}}}\cosh x=\cosh x\,.}

All functions with this property arelinear combinations ofsinh andcosh, in particular theexponential functionsex{\displaystyle e^{x}} ande−x{\displaystyle e^{-x}}.[24]

Standard integrals

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

∫sinh⁡(ax)dx=a−1cosh⁡(ax)+C∫cosh⁡(ax)dx=a−1sinh⁡(ax)+C∫tanh⁡(ax)dx=a−1ln⁡(cosh⁡(ax))+C∫coth⁡(ax)dx=a−1ln⁡|sinh⁡(ax)|+C∫sech⁡(ax)dx=a−1arctan⁡(sinh⁡(ax))+C∫csch⁡(ax)dx=a−1ln⁡|tanh⁡(ax2)|+C=a−1ln⁡|coth⁡(ax)−csch⁡(ax)|+C=−a−1arcoth⁡(cosh⁡(ax))+C{\displaystyle {\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:∫1a2+u2du=arsinh⁡(ua)+C∫1u2−a2du=sgn⁡uarcosh⁡|ua|+C∫1a2−u2du=a−1artanh⁡(ua)+Cu2<a2∫1a2−u2du=a−1arcoth⁡(ua)+Cu2>a2∫1ua2−u2du=−a−1arsech⁡|ua|+C∫1ua2+u2du=−a−1arcsch⁡|ua|+C{\displaystyle {\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}}}

whereC is theconstant of integration.

Taylor series expressions

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

sinh⁡x=x+x33!+x55!+x77!+⋯=∑n=0∞x2n+1(2n+1)!{\displaystyle \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 isconvergent for everycomplex value ofx. Since the functionsinhx isodd, only odd exponents forx occur in its Taylor series.

cosh⁡x=1+x22!+x44!+x66!+⋯=∑n=0∞x2n(2n)!{\displaystyle \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 isconvergent for everycomplex value ofx. Since the functioncoshx iseven, only even exponents forx occur in its Taylor series.

The sum of the sinh and cosh series is theinfinite series expression 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.tanh⁡x=x−x33+2x515−17x7315+⋯=∑n=1∞22n(22n−1)B2nx2n−1(2n)!,|x|<π2coth⁡x=x−1+x3−x345+2x5945+⋯=∑n=0∞22nB2nx2n−1(2n)!,0<|x|<πsech⁡x=1−x22+5x424−61x6720+⋯=∑n=0∞E2nx2n(2n)!,|x|<π2csch⁡x=x−1−x6+7x3360−31x515120+⋯=∑n=0∞2(1−22n−1)B2nx2n−1(2n)!,0<|x|<π{\displaystyle {\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:

Infinite products and continued fractions

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The following expansions are valid in the whole complex plane:

sinh⁡x=x∏n=1∞(1+x2n2π2)=x1−x22⋅3+x2−2⋅3x24⋅5+x2−4⋅5x26⋅7+x2−⋱{\displaystyle \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=∏n=1∞(1+x2(n−1/2)2π2)=11−x21⋅2+x2−1⋅2x23⋅4+x2−3⋅4x25⋅6+x2−⋱{\displaystyle \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=11x+13x+15x+17x+⋱{\displaystyle \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

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

The hyperbolic functions represent an expansion oftrigonometry beyond thecircular functions. Both types depend on anargument, eithercircular angle orhyperbolic angle.

Since thearea of a circular sector with radiusr and angleu (in radians) isr2u/2, it will be equal tou whenr =√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 sector with area corresponding to hyperbolic angle magnitude.

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

The hyperbolic angle is aninvariant measure with respect to thesqueeze mapping, just as the circular angle is invariant under rotation.[25]

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

The graph of the functiona cosh(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

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The decomposition of the exponential function in itseven and odd parts gives the identitiesex=cosh⁡x+sinh⁡x,{\displaystyle e^{x}=\cosh x+\sinh x,}ande−x=cosh⁡x−sinh⁡x.{\displaystyle e^{-x}=\cosh x-\sinh x.}Combined withEuler's formulaeix=cos⁡x+isin⁡x,{\displaystyle e^{ix}=\cos x+i\sin x,}this givesex+iy=(cosh⁡x+sinh⁡x)(cos⁡y+isin⁡y){\displaystyle e^{x+iy}=(\cosh x+\sinh x)(\cos y+i\sin y)}for thegeneral complex exponential function.

Additionally,ex=1+tanh⁡x1−tanh⁡x=1+tanh⁡x21−tanh⁡x2{\displaystyle 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

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Hyperbolic functions in the complex plane
sinh⁡(z){\displaystyle \sinh(z)}cosh⁡(z){\displaystyle \cosh(z)}tanh⁡(z){\displaystyle \tanh(z)}coth⁡(z){\displaystyle \coth(z)}sech⁡(z){\displaystyle \operatorname {sech} (z)}csch⁡(z){\displaystyle \operatorname {csch} (z)}

Since theexponential function can be defined for anycomplex argument, we can also extend the definitions of the hyperbolic functions to complex arguments. The functionssinhz andcoshz are thenholomorphic.

Relationships to ordinary trigonometric functions are given byEuler's formula for complex numbers:eix=cos⁡x+isin⁡xe−ix=cos⁡x−isin⁡x{\displaystyle {\begin{aligned}e^{ix}&=\cos x+i\sin x\\e^{-ix}&=\cos x-i\sin x\end{aligned}}}so:cosh⁡(ix)=12(eix+e−ix)=cos⁡xsinh⁡(ix)=12(eix−e−ix)=isin⁡xcosh⁡(x+iy)=cosh⁡(x)cos⁡(y)+isinh⁡(x)sin⁡(y)sinh⁡(x+iy)=sinh⁡(x)cos⁡(y)+icosh⁡(x)sin⁡(y)tanh⁡(ix)=itan⁡xcosh⁡x=cos⁡(ix)sinh⁡x=−isin⁡(ix)tanh⁡x=−itan⁡(ix){\displaystyle {\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 areperiodic with respect to the imaginary component, with period2πi{\displaystyle 2\pi i} (πi{\displaystyle \pi i} for hyperbolic tangent and cotangent).

See also

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References

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  1. ^abcdWeisstein, Eric W."Hyperbolic Functions".mathworld.wolfram.com. Retrieved2020-08-29.
  2. ^(1999)Collins Concise Dictionary, 4th edition, HarperCollins, Glasgow,ISBN 0 00 472257 4, p. 1386
  3. ^abCollins Concise Dictionary, p. 328
  4. ^ab"Hyperbolic Functions".www.mathsisfun.com. Retrieved2020-08-29.
  5. ^Collins Concise Dictionary, p. 1520
  6. ^Collins Concise Dictionary, p. 329
  7. ^tanh
  8. ^Collins Concise Dictionary, p. 1340
  9. ^Woodhouse, N. M. J. (2003),Special Relativity, London: Springer, p. 71,ISBN 978-1-85233-426-0
  10. ^Abramowitz, Milton;Stegun, Irene A., eds. (1972),Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York:Dover Publications,ISBN 978-0-486-61272-0
  11. ^Some examples of usingarcsinh found inGoogle Books.
  12. ^Niven, Ivan (1985).Irrational Numbers. Vol. 11. Mathematical Association of America.ISBN 9780883850381.JSTOR 10.4169/j.ctt5hh8zn.
  13. ^abGeorge F. Becker; C. E. Van Orstrand (1909).Hyperbolic Functions. Universal Digital Library. The Smithsonian Institution.
  14. ^abcdefMcMahon, James (1896).Hyperbolic Functions. Osmania University, Digital Library Of India. John Wiley And Sons.
  15. ^abcdBradley, Robert E.; D'Antonio, Lawrence A.; Sandifer, Charles Edward.Euler at 300: an appreciation. Mathematical Association of America, 2007. Page 100.
  16. ^Becker, Georg F.Hyperbolic functions. Read Books, 1931. Page xlviii.
  17. ^N.P., Bali (2005).Golden Integral Calculus. Firewall Media. p. 472.ISBN 81-7008-169-6.
  18. ^Steeb, Willi-Hans (2005).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 (3rd ed.). World Scientific Publishing Company. p. 281.ISBN 978-981-310-648-2.Extract of page 281 (using lambda=1)
  19. ^Oldham, Keith B.; Myland, Jan; Spanier, Jerome (2010).An Atlas of Functions: with Equator, the Atlas Function Calculator (2nd, illustrated ed.). Springer Science & Business Media. p. 290.ISBN 978-0-387-48807-3.Extract of page 290
  20. ^Osborn, G. (July 1902)."Mnemonic for hyperbolic formulae".The Mathematical Gazette.2 (34): 189.doi:10.2307/3602492.JSTOR 3602492.S2CID 125866575.
  21. ^Martin, George E. (1986).The foundations of geometry and the non-Euclidean plane (1st corr. ed.). New York: Springer-Verlag. p. 416.ISBN 3-540-90694-0.
  22. ^"Prove the identity tanh(x/2) = (cosh(x) - 1)/sinh(x)".StackExchange (mathematics). Retrieved24 January 2016.
  23. ^Audibert, Jean-Yves (2009). "Fast learning rates in statistical inference through aggregation". The Annals of Statistics. p. 1627.[1]
  24. ^Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010),"Hyperbolic functions",NIST Handbook of Mathematical Functions, Cambridge University Press,ISBN 978-0-521-19225-5,MR 2723248.
  25. ^Haskell, Mellen W., "On the introduction of the notion of hyperbolic functions",Bulletin of the American Mathematical Society1:6:155–9,full text

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