Introductions To Sech H L H L H L H L H L H L

http://functions.wolfram.com7The attractive feature of the Poincaré disk model is that the hyperbolic angles agree with the Euclidean angles.Formally, the angle Α at a point A of two hyperbolic lines A B and A C is described by the formula:cosHΑL -A B -A C1-A B 1-A C 1-A B 1-A C -A B-A C.In the following, the values of the three angles of an hyperbolic triangle at the vertices A, B, and C are denotedthrough Α, Β, and Γ. The hyperbolic length of the three edges opposite to the angles are denoted a, b, and c.The cosine rule and the second cosine rule for hyperbolic triangles are:sinhHbL sinhHcL cosHΑL coshHbL coshHcL - coshHaLsinhHaL sinhHcL cosHΒL coshHaL coshHcL - coshHbLsinhHaL sinhHbL cosHΓL coshHaL coshHbL - coshHcLsinHΒL sinHΓL coshHaL cosHΒL cosHΓL cosHΑLsinHΑL sinHΓL coshHbL cosHΑL cosHΑL cosHΒLsinHΑL sinHΒL coshHcL cosHΑL cosHΒL cosHΓL.The sine rule for hyperbolic triangles is:sinHΑLsinHΒL sinhHaL sinhHbLsinHΓL.sinhHcLFor a right-angle triangle, the hyperbolic version of the Pythagorean theorem follows from the preceding formulas(the right angle is taken at vertex A):coshHaL coshHbL coshHcL.Using the series expansion coshHxL » 1 x2 2 at small scales the hyperbolic geometry is approximated by thefamilar Euclidean geometry. The cosine formulas and the sine formulas for hyperbolic triangles with a right angleat vertex A become:cosHΒL tanhHcLsinhHbL, sinHΒL tanhHaLtanhHbLcosHΓL sinhHaLsinhhHcL, sinHΓL .tanhHaLtanhHaLThe inscribed circle has the radius:Ρ -1tanhcos2 HΑL cos2 HΒL cos2 HΓL 2 cosHΑL cosHΒL cosHΓL - 12 H1 cosHΑLL H1 cosHΒLL H1 cosHΓLLThe circumscribed circle has the radius:4 sinhI 2 M sinhJ 2 N sinhI 2 Ma-1Ρ tanhbsinHΓL sinhHaL sinhHbLc.

http://functions.wolfram.com8Other applicationsAs rational functions of the exponential function, the hyperbolic functions appear virtually everywhere in quantitative sciences. It is impossible to list their numerous applications in teaching, science, engineering, and art.Introduction to the Hyperbolic Secant FunctionDefining the hyperbolic secant functionThe hyperbolic secant function is an old mathematical function.This function is easily defined as the ratio of one and hyperbolic cosine functions:1sechHzL coshHzL2ã ã-zz.After comparison with the famous Euler formula for the cosine function cosHzL ãä z ã-ä z,2it is easy to derive thefollowing representation of the hyperbolic secant through the circular secant:sechHzL secHä zL.The previous formula allows establishment of all the properties and formulas for the hyperbolic secant fromcorresponding properties and formulas for the circular secant.A quick look at the hyperbolic secant functionHere is a graphic of the hyperbolic secant function f HxL sechHxL for real values of its argument x.21.5f10.50-0.5-1-2-10x12Representation through more general functionsThe hyperbolic secant function sechHzL can be represented using more general mathematical functions. As thereciprocal to the hyperbolic cosine function that is a particular case of the generalized hypergeometric, Bessel,Struve, and Mathieu functions, the hyperbolic secant function can also be represented as reciprocal to those specialfunctions. Here are some examples:sech HzL 10 F11 z2; ;2 4sech HzL 1Πäz2J-12Hä zLsech HzL 1Πz2I-12HzL

http://functions.wolfram.comsech HzL -91Πäz2Y 1 Hä zLsech HzL 1Πz1 2L 1 HzLsech HzL 1.Ce H1,0,ä zL22But these representations are not very useful because they include complicated special functions in thedenominators.It is more useful to write the secant function as particular cases of one special function. That can be done usingdoubly periodic Jacobi elliptic functions that degenerate into the hyperbolic secant function when their secondparameter is equal to 0 or 1:sechHzL dcHä z È 0L ncHä z È 0L dsΠ2- ä z 0 nsΠ2- ä z 0 cnHz È 1L dnHz È 1L ä csΠä2- z 1 ä dsΠä2-z 1 .Definition of the hyperbolic secant function for a complex argumentIn the complex z-plane, the function sechHzL is defined by the same formula used for real values:1sechHzL coshHzL2ã ã-zz.In the points z Π ä 2 Π k ä ; k Î Z, where coshHzL has zeros, the denominator of the last formula equals zero andsechHzL has singularities (poles of the first order).Here are two graphics showing the real and imaginary parts of the hyperbolic secant function over the complexplane.2Re2Im02-202-20-20-2x2y0-20y-2x2The best-known properties and formulas for the hyperbolic secant functionValues in pointsThe values of the hyperbolic secant function for special values of its argument can be easily derived from thecorresponding values of the circular secant function in special points of the circle:

http://functions.wolfram.comsech H0L 1sech JΠäN2 sech HΠ äL -1sech J3ΠäN210sech Jsech JΠäN62ΠäN37Πäsech J 6 N sech Jsech J2 sech J3 -2 -5ΠäN323ΠäN43ΠäN45Πäsech J 4 N 2sech J 2 - 2 - 27ΠäN4 2sech Jsech JΠäN35ΠäN6 -4Πäsech J 3 Nsech J 211 Π äN623 -2 231 ; m Î Z.sech H2 Π äL 1 sech HΠ ä mL H-1Lm ; m Î Z sech JΠ ä J 2 mNN The values at infinity can be expressed by the following formulas:sech H L 0 sech H- L 0.General characteristicsFor real values of argument z, the values of sechHzL are real.In the points z 2 Π n ä m ; n Î Z ì m Î Z, the values of sechHzL are algebraic. In several cases, they can beintegers -2, -1, 1, or 2:sech H0L 1 sech JΠäN3The values of sechI 2 sech JΠäΠMm2ΠäN3 -2 sech HΠ äL -1.can be expressed using only square roots if n Î Z and m is a product of a power of 2 anddistinct Fermat primes {3, 5, 17, 257, }.The function sechHzL is an analytical function of z that is defined over the whole complex z-plane and does not havebranch cuts and branch points. It has an infinite set of singular points:(a) z Π ä 2 Π ä k ; k Î Z are the simple poles with residues H-1Lk-1 ä. (b) z is an essential singular point.It is a periodic function with period 2 Π ä:sechHz 2 Π äL sechHzLsech HzL sech Hz 2 Π ä kL ; k Î Z sech HzL H-1Lk sech Hz Π ä kL ; k Î Z.The function sechHzL is an even function with mirror symmetry:sech H-zL sech HzL sech Hz L sech HzL .DifferentiationThe first derivative of sechHzL has simple representations using either the tanhHzL function or the sechHzL function:¶sech HzL¶z -tanh HzL sech HzL.

http://functions.wolfram.com11The nth derivative of sechHzL has much more complicated representations than the symbolic nth derivatives forsinhHzL and coshHzL:¶n sech HzL¶znn fvk-1 sech HzL n ä Hn 1L! â â2nIH-1Lk 21-k Hk - 2 jLn sechk HzLM cosh JänΠ2Hk 1L j ! Hk - jL! Hn - kL!k 0 j 0- Hk - 2 jL zN ; n Î N,where n is the Kronecker delta symbol: 0 1 and n 0 ; n ¹ 0.Ordinary differential equationThe functions sechHzL satisfies the following first-order nonlinear differential equation:w HzL2 w HzL4 - w HzL2 0 ; w HzL sech HzL.Series representationThe function sechHzL has the following series expansion at the origin that converges for all finite values z withΠz 2 :sech HzL 1 -z2 25 z424 ¼ â E2 k z2 kH2 kL!k 0,where E2 k are the Euler numbers.The hyperbolic secant function sechHzL can also be presented using other kinds of series with the followingformulas:sech HzL Π âH-1Lk H2 k 1L k 0Π Jk 2sech HzL - â1 2N2 ; z212k - Jz Π ä Jk 2 NN21äzΠ ;-12äzΠÏZ-12Ï Z.Integral representationThe function sechHzL has a well-known integral representation through the following definite integral along thepositive part of the real axis:sechHzL 2Πà0 1t 12tâ t ; ImHzL 2äzΠΠ2.Product representationThe famous infinite product representation for coshHzL can be easily rewritten as the following product representation for the hyperbolic secant function:sechHzL ä k 1Π2 H2 k - 1L2Π2 H2 k - 1L 4 z22.

http://functions.wolfram.com12Limit representationThe hyperbolic secant function has the following limit representation:H-1Lksech HzL lim ânn Π Jk 2 N - ä z1k -n ;äzΠ-12Ï Z.Indefinite integrationIndefinite integrals of expressions that contain the hyperbolic secant function can sometimes be expressed usingelementary functions. However, special functions are frequently needed to express the results even when theintegrands have a simple form (if they can be evaluated in closed form). Here are some examples:z-1à sechHzL â z 2 tan KtanhK OO2sechHzL â z -2 ä cosh 2 HzL Fà1và sech Ha zL â z -v-1secha H1 - vLäz22 sech 2 HzL1Ha zL sinh Ha zL-sinh Ha zL22 F11-v 1 3-v2, ;; cosh Ha zL .222Definite integrationDefinite integrals that contain the hyperbolic secant function are sometimes simple and their values can beexpressed through elementary functions. Here is one example:à t2 sechHtL â t 0Π3.8Some special functions can be used to evaluate more complicated definite integrals. For example, gamma functions, incomplete beta functions, and the Catalan constant are needed to express the following integrals:àsech HtL â t Π2a0à12ä Bcosh2 J Π N21-a 1,22Π GJ2 - 2N1aG I1 - 2 Ma ; Re HaL 1 t sechHtL â t 2 C.0Finite summationThe following finite sum that contain the hyperbolic secant has a simple value:ânk 1122k2zsech2k1 22ncsch K2z2nO - csch HzL ; n Î N.2Infinite summationThe evaluation limit of the last formula in the previous subsubsection for n - gives the following value for thecorresponding infinite sum:

http://functions.wolfram.comâ k 1122 kz2sech2k 131z2- csch HzL.2Finite productsThe following finite product from the hyperbolic secant can be represented through the hyperbolic cosecantfunction:ä sec z n-1k 0Πkn H-1Ln-1 2n-1 cscn HΠ - 2 zL ; n Î N .2Infinite productsThe following infinite product from the hyperbolic secant can be represented through the hyperbolic cosecantfunction:ä sech zk 12k z cschHzL.Addition formulasThe hyperbolic secant of a sum or difference can be represented in terms of hyperbolic sine and cosine as shown inthe following formulas:sechHa bL sechHa - bL 1coshHaL coshHbL sinhHaL sinhHbL1coshHbL coshHaL - sinhHaL sinhHbL.Multiple argumentsIn the case of multiple arguments 2 z, 3 z, , the function sechHn zL can be represented as a rational function thatcontains powers of the hyperbolic secant. Here are two examples:sech HzL2sechH2 zL 2 - sech HzL2sech HzL3sechH3 zL 4 - 3 sech HzL2.Half-angle formulasThe hyperbolic secant of a half-angle can be represented by the following simple formula that is valid in a horizontal strip:zsechK O 22coshHzL 1 ; ImHzL Π Þ HImHzL Π ß ReHzL 0L Þ HImHzL -Π ß ReHzL 0L.To make this formula correct for all complex z, a complicated prefactor is needed:

http://functions.wolfram.comzsechK O cHzL214 ; cHzL H-1Lf2coshHzL 1Π-ImHzL2Πv1 - 1 H-1LfΠ-ImHzL2Πv f-Π-ImHzL2ΠvΘH-ReHzLL ,where cHzL contains the unit step, real part, imaginary part, and the floor functions.Sums of two direct functionsThe sum and difference of two hyperbolic secant functions can be described by the following formulas:asechHaL sechHbL 2 cosh2sechHaL - sechHbL -2 sinha2bacosh2-b2 2asinh2b2 sechHaL sechHbLb2sechHaL sechHbL.Products involving the direct functionThe product of two hyperbolic secants and the product of hyperbolic secant and cosecant have the followingrepresentations:2sechHaL sechHbL coshHa - bL coshHa bL2sechHaL cschHbL .sinhHa bL - sinhHa - bLInequalitiesOne of the most famous inequalities for the hyperbolic secant function is the following:sech HxL x csch HxL ; x 0 ß x Î R.Relations with its inverse functionThere are simple relations between the function sechHzL and its inverse function sech-1 HzL:sech Isech-1HzLM z sech-1Hsech HzLL z ; -Π Im HzL Π ß Re HzL 0 Þ Re HzL 0 ß Im HzL ³ 0.The second formula is valid at least in the horizontal half-strip -Π ImHzL Π ì ReHzL 0. This can be expandedto a full horizontal strip:sech HsechHzLL -1z2 ; -Π ImHzL Π Þ ImHzL -Π ß ReHzL 0 Þ ImHzL Π ß ReHzL ³ 0.Outside of this strip a much more complicated relation (that contains the unit step, real part, and the floor functions)holds:sech HsechHzLL -1z2 1 -Πä2z1 - H-1LfImHzLvΠ 2ImHzLΠRepresentations through other hyperbolic functions Πä2z2z 1 1 H-1LfImHzL2Π v f1ImHzL22Π- v12ΘHReHzLL ; z ¹ 0.

http://functions.wolfram.com15The hyperbolic secant and cosecant functions are connected by a very simple formula including the linear functionin the argument:sechHzL ä cschΠä2-z .The hyperbolic secant function can also be represented using other hyperbolic functions by the following formulas:sech HzL äsinh JΠä2-zNsech HzL 1-tanh2 J Nz2z1 tanh2 J N2sech HzL coth2 J N-1z2zcoth2 J N 12.Representations through trigonometric functionsThe hyperbolic secant function has the following representations using the trigonometric functions:sech HzL 1sin J -ä zNΠsech HzL 1cos Hä zLsech HzL 21 tan2 JäzN2äz1-tan2 J N2sech HzL sech HzL csc I 2 - ä zM sech HzL sec Hä zL sech Hä zL sec HzL.cot2 JäzN 12äzcot2 J N-12ΠApplicationsThe hyperbolic secant function is used throughout mathematics, the exact sciences, and engineering.Introduction to the Hyperbolic Functions in MathematicaOverviewThe following shows how the six hyperbolic functions are realized in Mathematica. Examples of evaluatingMathematica functions applied to various numeric and exact expressions that involve the hyperbolic functions orreturn them are shown. These involve numeric and symbolic calculations and plots.NotationsMathematica forms of notationsAll six hyperbolic functions are represented as built-in functions in Mathematica. Following Mathematica's generalnaming convention, the StandardForm function names are simply capitalized versions of the traditional mathematics names. Here is a list hypFunctions of the six hyperbolic functions in StandardForm.hypFunctions 8Sinh@zD, Cosh@zD, Tanh@zD, Coth@zD, Sech@zD, Cosh@zD 8Sinh@zD, Cosh@zD, Tanh@zD, Coth@zD, Sech@zD, Cosh@zD Here is a list hypFunctions of the six trigonometric functions in TraditionalForm.hypFunctions TraditionalForm8sinhHzL, coshHzL, tanhHzL, cothHzL, sechHzL, coshHzL Additional forms of notations

http://functions.wolfram.com16Mathematica also knows the most popular forms of notations for the hyperbolic functions that are used in otherprogramming languages. Here are three examples: CForm, TeXForm, and FortranForm.hypFunctions . 8z 2 Π z oth(2*Pi*z),Sech(2*Pi*z),Cosh(2*Pi*z))hypFunctions . 8z 2 Π z TeXForm\{ \sinh (2\,\pi \,z),\cosh (2\,\pi \,z),\tanh (2\,\pi \,z),\coth (2\,\pi \,z),\Mfunction{Sech}(2\,\pi \,z),\cosh (2\,\pi \,z)\}hypFunctions . 8z 2 Π z tic evaluations and transformationsEvaluation for exact, machine-number, and high-precision argumentsFor a simple exact argument, Mathematica returns an exact result. For instance, for the argument Π ä 6, the Sinhfunction evaluates to ä 2.FΠäSinhB6ä28Sinh@zD, Cosh@zD, Tanh@zD, Coth@zD, Csch@zD, Sech@zD . z ä: ,2ä3,2, -ä3 , -2 ä,32Πä6 3For a generic machine-number argument (a numerical argument with a decimal point and not too many digits), amachine number is returned.Cosh@3.D10.06778Sinh@zD, Cosh@zD, Tanh@zD, Coth@zD, Csch@zD, Sech@zD . z 2.83.62686, 3.7622, 0.964028, 1.03731, 0.275721, 0.265802 The next inputs calculate 100-digit approximations of the six hyperbolic functions at z 1.N@Tanh@1D, 1D N@ð, 50D 05

http://functions.wolfram.com17N@8Sinh@zD, Cosh@zD, Tanh@zD, Coth@zD, Csch@zD, Sech@zD . z 1, 81334095870229565413013307567304323895 98 83 382 03 830 2808547853078928924 Within a second, it is possible to calculate thousands of digits for the hyperbolic functions. The next input calculates 10000 digits for sinhH1L, coshH1L, tanhH1L, cothH1L, sechH1L, and cschH1L and analyzes the frequency of theoccurrence of the digit k in the resulting decimal number.Map@Function@w, 8First@ðD, Length@ðD & Split@Sort@First@RealDigits@wDDDDD,N@8Sinh@zD, Cosh@zD, Tanh@zD, Coth@zD, Csch@zD, Sech@zD . z 1, 10 000DD8880,86,83,880,87,83,880,87,83,980 , 81, 994 , 82, 996 , 83, 1014 , 84, 986 , 85, 1001 ,1017 , 87, 1020 , 88, 981 , 89, 1011 , 880, 1015 , 81, 960 , 82, 997 ,1037 , 84, 1070 , 85, 1018 , 86, 973 , 87, 997 , 88, 963 , 89, 970 ,971 , 81, 1023 , 82, 1016 , 83, 970 , 84, 949 , 85, 1052 , 86, 981 ,1056 , 88, 1010 , 89, 972 , 880, 975 , 81, 986 , 82, 1023 ,1004 , 84, 1008 , 85, 977 , 86, 977 , 87, 1036 , 88, 1035 , 89, 979 ,979 , 81, 1030 , 82, 987 , 83, 992 , 84, 1016 , 85, 1030 , 86, 1021 ,969 , 88, 974 , 89, 1002 , 880, 1009 , 81, 971 , 82, 1018 ,994 , 84, 1011 , 85, 1018 , 86, 958 , 87, 1019 , 88, 1016 , 89, 986 Here are 50-digit approximations to the six hyperbolic functions at the complex argument z 3 5 ä.N@Csch@3 5 äD, 23887285931631736730964453318082730911 1484269546408531396 775479803879772793331583262276221 38939784445056701747 äN@8Sinh@zD, Cosh@zD, Tanh@zD, Coth@zD, Csch@zD, Sech@zD . z 3 5 ä, 50D

5515436340301659921919691213853 94033 59125751 04102 1108 388729 775 614624 833 ä Mathematica always evaluat

csch2 HzL 1 sinh2 HzL sech HzL 1 cosh HzL ä sinh J pä 2-zN sech2 HzL 1 1 sinh2 HzL. All six hyperbolic functions can be transformed into any other function of the group of hyperbolic functions if the argument z is replaced by ppä’2 qz with q2 1ìp Z: sinh H-z-2päL -sinh HzLsinh Hz-