Hyperbolic function 1

Hyperbolic function

A ray through the origin intercepts the unit hyperbola in the point, where is twice the area between the ray, the hyperbola, and

the -axis. For points on the hyperbola below the -axis, the area is considerednegative (see animated version with comparison with the trigonometric (circular)

functions).

In mathematics, hyperbolic functions areanalogs of the ordinary trigonometric, orcircular, functions. The basic hyperbolicfunctions are the hyperbolic sine "sinh"(/snt/ or /an/), and the hyperbolic cosine"cosh" /k/, from which are derived thehyperbolic tangent "tanh" (/tnt/ or/n/), hyperbolic cosecant "csch" or"cosech" /kok/ or /kost/, hyperbolicsecant "sech" /k/ or /st/, andhyperbolic cotangent "coth" /ko/ or/k/,[1] corresponding to the derivedtrigonometric functions. The inversehyperbolic functions are the area hyperbolicsine "arsinh" (also called "asinh" orsometimes "arcsinh")[2] and so on.

Just as the points (cost,sint) form a circlewith a unit radius, the points (cosht,sinht)form the right half of the equilateralhyperbola. Hyperbolic functions occur in thesolutions of some important linear differential equations, for example the equation defining a catenary, of somecubic equations, and of Laplace's equation in Cartesian coordinates. The latter is important in many areas of physics,including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.

The hyperbolic functions take real values for a real argument called a hyperbolic angle. The size of a hyperbolicangle is the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a righttriangle covering this sector.

In complex analysis, the hyperbolic functions arise as the imaginary parts of sine and cosine. When considereddefined by a complex variable, the hyperbolic functions are rational functions of exponentials, and are hencemeromorphic.

Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann HeinrichLambert.[3] Riccati used Sc. and Cc. ([co]sinus circulare) to refer to circular functions and Sh. and Ch. ([co]sinushyperbolico) to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what theyare today.[4] The abbreviations sh and ch are still used in some other languages, like European French and Russian.

Hyperbolic function 2

Standard algebraic expressions

sinh, cosh and tanh

csch, sech and coth

(a) cosh(x) is the average of exand ex

(b) sinh(x) is half the difference of ex and ex

Hyperbolic functions (a) cosh and (b) sinh obtained using exponential functions and The hyperbolic functions are: Hyperbolic sine:

Hyperbolic cosine:

Hyperbolic tangent:

Hyperbolic cotangent:

Hyperbolic function 3

Hyperbolic secant:

Hyperbolic cosecant:

Hyperbolic functions can be introduced via imaginary circular angles: Hyperbolic sine:

Hyperbolic cosine:

Hyperbolic tangent:

Hyperbolic cotangent:

Hyperbolic secant:

Hyperbolic cosecant:

where i is the imaginary unit defined by i2 = 1.The complex forms in the definitions above derive from Euler's formula.

Useful relationsOdd and even functions:

Hence:

It can be seen that cosh x and sech x are even functions; the others are odd functions.

Hyperbolic sine and cosine satisfy the identity

Hyperbolic function 4

which is similar to the Pythagorean trigonometric identity. One also has

for the other functions.The hyperbolic tangent is the solution to the nonlinear boundary value problem:

It can be shown that the area under the curve of cosh(x) is always equal to the arc length:[5]

Sums of arguments:

particularly

Sum and difference of cosh and sinh:

Inverse functions as logarithms

Hyperbolic function 5

Derivatives

Standard integralsFor a full list of integrals of hyperbolic functions, see list of integrals of hyperbolic functions.

Hyperbolic function 6

where C is the constant of integration.

Taylor series expressionsIt is possible to express the above functions as Taylor series:

The function sinhx has a Taylor series expression with only odd exponents for x. Thus it is an odd function, that is,sinhx=sinh(x), and sinh0=0.

The function coshx has a Taylor series expression with only even exponents for x. Thus it is an even function, thatis, symmetric with respect to the y-axis. The sum of the sinh and cosh series is the infinite series expression of theexponential function.

whereis the nth Bernoulli numberis the nth Euler number

Hyperbolic function 7

Comparison with circular functions

Circle and hyperbola tangent at (1,1) display geometry of circularfunctions in terms of circular sector area u and hyperbolic functions

depending on hyperbolic sector area u.

The hyperbolic functions represent an expansion oftrigonometry beyond the circular functions. Both typesdepend on an argument, either circular angle orhyperbolic angle.

Since the area of a circular sector is it will be

equal to u when r = square root of 2. In the diagramsuch a circle is tangent to the hyperbola x y = 1 at (1,1).The yellow sector depicts an area and angle magnitude.Similarly, the red augmentation depicts an area andmagnitude as hyperbolic angle.The legs of the two right triangles with hypotenuse onthe ray defining the angles are of length 2 times thecircular and hyperbolic functions.

Identities

The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact,Osborn's rule[6] states that one can convert any trigonometric identity into a hyperbolic identity by expanding itcompletely in terms of integral powers of sines and cosines, changing sine to sinh and cosine to cosh, and switchingthe sign of every term which contains a product of 2, 6, 10, 14, ... sinhs. This yields for example the additiontheorems

the "double argument formulas"

and the "half-argument formulas"[7]

Note: This is equivalent to its circular counterpart multiplied by 1.

Note: This corresponds to its circular counterpart.

Hyperbolic function 8

The derivative of sinhx is coshx and the derivative of coshx is sinhx; this is similar to trigonometric functions,albeit the sign is different (i.e., the derivative of cosx is sinx).The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic ones thatdoes not involve complex numbers.The graph of the function acosh(x/a) is the catenary, the curve formed by a uniform flexible chain hanging freelybetween two fixed points under uniform gravity.

Relationship to the exponential functionFrom the definitions of the hyperbolic sine and cosine, we can derive the following identities:

and

These expressions are analogous to the expressions for sine and cosine, based on Euler's formula, as sums ofcomplex exponentials.

Hyperbolic functions for complex numbersSince the exponential function can be defined for any complex argument, we can extend the definitions of thehyperbolic functions also to complex arguments. The functions sinhz and coshz are then holomorphic.Relationships to ordinary trigonometric functions are given by Euler's formula for complex numbers:

so:

Thus, hyperbolic functions are periodic with respect to the imaginary component, with period ( forhyperbolic tangent and cotangent).

Hyperbolic function 9

Hyperbolic functions in the complex plane

References[1] tanh (http:/ / www. mathcentre. ac. uk/ resources/ workbooks/ mathcentre/ hyperbolicfunctions. pdf)[2] Some examples of using arcsinh (http:/ / www. google. com/ books?q=arcsinh+ -library) found in Google Books.[3] Robert E. Bradley, Lawrence A. D'Antonio, Charles Edward Sandifer. Euler at 300: an appreciation. Mathematical Association of America,

2007. Page 100.[4] Georg F. Becker. Hyperbolic functions. Read Books, 1931. Page xlviii.[5] , Extract of page 472 (http:/ / books. google. com/ books?id=hfi2bn2Ly4cC& pg=PA472)[6] G. Osborn, Mnemonic for hyperbolic formulae (http:/ / links. jstor. org/ sici?sici=0025-5572(190207)2:2:342. 0. CO;2-Z), The

Mathematical Gazette, p. 189, volume 2, issue 34, July 1902[7] , Chapter 26, page 1155 (http:/ / books. google. com/ books?id=PGuSDjHvircC& pg=PA1155)

External links Hazewinkel, Michiel, ed. (2001), "Hyperbolic functions" (http:/ / www. encyclopediaofmath. org/ index.

php?title=p/ h048250), Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4 Hyperbolic functions (http:/ / planetmath. org/ encyclopedia/ HyperbolicFunctions. html) on PlanetMath Hyperbolic functions (http:/ / mathworld. wolfram. com/ HyperbolicFunctions. html) entry at MathWorld GonioLab (http:/ / glab. trixon. se/ ): Visualization of the unit circle, trigonometric and hyperbolic functions (Java

Web Start) Web-based calculator of hyperbolic functions (http:/ / www. calctool. org/ CALC/ math/ trigonometry/

hyperbolic)

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Article Sources and ContributorsHyperbolic function Source: http://en.wikipedia.org/w/index.php?oldid=604474202 Contributors: A876, Aaron Rotenberg, Access Denied, Agro r, AlanUS, Albmont, Alle, Allispaul,AndrewWTaylor, Andy Fisher aKa Shine, Armeria wiki, Ashis Kumar Sahoo, AxelBoldt, Azo bob, Barak Sh, Bcherkas, Ben pcc, BenFrantzDale, Bh3u4m, Bhadani, Blah314, Brad7777,Brighterorange, Calvinballing, Can't sleep, clown will eat me, Canderson7, Carlodn6, Charles Matthews, Chris-gore, ChrisGualtieri, Chzz, CiaPan, Coasterlover1994, Cojoco, Compotatoj,Cottonmouse, Courcelles, Ctifumdope, Culero101, Cyp, DVdm, Daniele Pugliesi, Dcqec111, Deathbeast, Deflective, Derepi, Deskana, Dlituiev, Doctormatt, Double sharp, Dr Greg, Dragnmn,Drilnoth, Duoduoduo, Eettttt, Egmontaz, Eleolar, Email4mobile, Eric119, Error792, Evil saltine, Fanix1128, Faramir1138, Footyfanatic3000, Fredrik, Gabbe, Gauge, Gco, GeiwTeol, Gene WardSmith, Georg Muntingh, Georgia guy, Geraldshields11, Gesslein, Giftlite, Goffrie, Gogo Dodo, Googl, Guardian of Light, Hashar, Hdt83, Heliomance, Herbee, HowiAuckland, Husond, Icairns,IgnorantArmies, JP.Martin-Flatin, Jackfork, Jdh8, Jeandavid54, Jleedev, JoeHillen, John of Reading, Josh Grosse, Jrgsampaio, Justin W Smith, K4vin, KarlJorgensen, KennethJ, Kiensvay,KnowledgeOfSelf, Krishnavedala, Kritikool99, Ktims, Kwamikagami, LOL, Lambiam, Lee, Linas, LokiClock, LucasVB, Lzur, Macrakis, MadGuy7023, Maksim-e, MarSch, MarkSweep,Mate2code, Materialscientist, Matt24, Me and, Mebden, Metacomet, Michael Hardy, Michael.YX.Wu, Michiganfan9000, Mleconte, Montparnasse, Moverington, Moxfyre, N4m3, Nascar1996,Nbarth, Ninly, Noerdosnum, Nonagonal Spider, Oleg Alexandrov, PAR, Patchouli, Pikalax, Pne, Pokajanje, Policron, Prophile, Qantas94Heavy, Qwertyus, Reciprocist, Rgdboer, Richnfg,Roberdin, Rocketrod1960, Rocky2009, SKvalen, Salih, Sam Derbyshire, Scientific29, SebastianHelm, Seraph85, Sergiacid, Sharonlees, Shen, Shubham10,000, Silly rabbit, Sjakkalle, Smack,Snevets, Societelibre, Sohale, Steve911411, Suisui, Sverdrup, Swetrix, TV4Fun, TYelliot, TakuyaMurata, The Anome, TheBendster, Tim bates, Tob, Tobias Bergemann, Tsemii, Txebixev,Urhixidur, V1adis1av, Vinograd19, Vizziee, Vt Zajac, WIKIWIZDOM, Wagnerjp25, Wavelength, Wereon, Wiccan Quagga, WikHead, WojciechSwiderski, Wwoods, Xanthoxyl, Zvika, 413anonymous edits

Image Sources, Licenses and ContributorsImage:Hyperbolic functions-2.svg Source: http://en.wikipedia.org/w/index.php?title=File:Hyperbolic_functions-2.svg License: Public Domain Contributors: Hyperbolic_functions.svg:Original uploader was Marco Polo at en.wikipedia derivative work: Jeandavid54 (talk)Image:sinh cosh tanh.svg Source: http://en.wikipedia.org/w/index.php?title=File:Sinh_cosh_tanh.svg License: Public Domain Contributors: Original uploader was Ktims at en.wikipediaImage:csch sech coth.svg Source: http://en.wikipedia.org/w/index.php?title=File:Csch_sech_coth.svg License: Public Domain Contributors: Original uploader was Ktims at en.wikipediafile:Hyperbolic and exponential; cosh.svg Source: http://en.wikipedia.org/w/index.php?title=File:Hyperbolic_and_exponential;_cosh.svg License: Creative Commons Attribution-Sharealike3.0 Contributors: Krishnavedalafile:Hyperbolic and exponential; sinh.svg Source: http://en.wikipedia.org/w/index.php?title=File:Hyperbolic_and_exponential;_sinh.svg License: Creative Commons Attribution-Sharealike3.0 Contributors: KrishnavedalaFile:Circular and hyperbolic angle.svg Source: http://en.wikipedia.org/w/index.php?title=File:Circular_and_hyperbolic_angle.svg License: Creative Commons Zero Contributors:User:MaschenImage:Complex Sinh.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Complex_Sinh.jpg License: Public Domain Contributors: Jan HomannImage:Complex Cosh.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Complex_Cosh.jpg License: Public Domain Contributors: Jan HomannImage:Complex Tanh.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Complex_Tanh.jpg License: Public Domain Contributors: Jan HomannImage:Complex Coth.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Complex_Coth.jpg License: Public Domain Contributors: Jan HomannImage:Complex Sech.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Complex_Sech.jpg License: Public Domain Contributors: Jan HomannImage:Complex Csch.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Complex_Csch.jpg License: Public Domain Contributors: Jan Homann

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Hyperbolic functionStandard algebraic expressionsUseful relationsInverse functions as logarithmsDerivativesStandard integralsTaylor series expressionsComparison with circular functionsIdentitiesRelationship to the exponential functionHyperbolic functions for complex numbersReferencesExternal links

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