Introductions to Sinh

Transcription

1 Introductions to Sinh Introduction to the hyperbolic functions General The six well-known hyperbolic functions are the hyperbolic sine sinh, hyperbolic cosine cosh, hyperbolic tangent tanh, hyperbolic cotangent coth, hyperbolic cosecant csch, and hyperbolic secant sech. They are among the most used elementary functions. The hyperbolic functions share many common properties and they have many properties and formulas that are similar to those of the trigonometric functions. Definitions of the hyperbolic functions All hyperbolic functions can be defined as simple rational functions of the exponential function of : sinh cosh tanh coth csch sech. The functions tanh, coth, csch, and sech can also be defined through the functions sinh and cosh using the following formulas: tanh sinh cosh coth cosh sinh csch sinh sech cosh. A quick look at the hyperbolic functions Here is a quick look at the graphics of the six hyperbolic functions along the real axis.

2 f x sinhx coshx tanhx cothx sechx cschx Connections within the group of hyperbolic functions and with other function groups Representations through more general functions The hyperbolic functions are particular cases of more general functions. Among these more general functions, four classes of special functions are of special relevance: Bessel, Jacobi, Mathieu, and hypergeometric functions. For example, sinh and cosh have the following representations through Bessel, Mathieu, and hypergeometric functions: sinh Π J sinh Π I sinh Π Y sinh Π K cosh Π J cosh Π I cosh Π Y cosh Π K sinh Se, 0, cosh Ce, 0, sinh 0 F ; 3 ; 4 cosh 0F ; ; 4. All hyperbolic functions can be represented as degenerate cases of the corresponding doubly periodic Jacobi elliptic functions when their second parameter is equal to 0 or : sinh sd 0 sn 0 sinh sc sd cosh cd 0 cn 0 cosh nc nd tanh sc 0 tanh sn coth cs 0 coth ns csch ds 0 ns 0 csch cs ds sech dc 0 nc 0 sech cn dn. Representations through related equivalent functions Each of the six hyperbolic functions can be represented through the corresponding trigonometric function: sinh sin sinh sin cosh cos cosh cos tanh tan tanh tan coth cot coth cot csch csc csch csc sech sec sech sec. Relations to inverse functions

3 3 Each of the six hyperbolic functions is connected with a corresponding inverse hyperbolic function by two formulas. One direction can be expressed through a simple formula, but the other direction is much more complicated because of the multivalued nature of the inverse function: sinhsinh sinh sinh ; Π Im Π Im Π Re 0 Im Π Re 0 coshcosh cosh cosh ; Re 0 Π Im Π Re 0 Im 0 tanhtanh cothcoth cschcsch tanh tanh ; Π Im Π Im Π Re 0 Im Π Re 0 coth coth ; Π Im Π Im Π Re 0 Im Π Re 0 csch csch ; Π Im Π Im Π Re 0 Im Π Re 0 sechsech sech sech ; Π Im Π Re 0 Re 0 Im 0. Representations through other hyperbolic functions Each of the six hyperbolic functions can be represented through any other function as a rational function of that function with a linear argument. For example, the hyperbolic sine can be representative as a group-defining function because the other five functions can be expressed as: cosh sinh Π cosh sinh tanh sinh cosh coth cosh sinh csch sinh sech cosh sinh sinh Π Π sinh sinh tanh sinh sinh coth sinh sinh csch sinh sinh Π sech sinh. All six hyperbolic functions can be transformed into any other function of the group of hyperbolic functions if the argument is replaced by p Π q with q p : sinh Π sinh sinh 3 Π sinh Π sinh sinh Π sinh Π sinh 3 Π cosh sinh cosh sinh Π sinh Π cosh sinh cosh sinh Π cosh Π sinh cosh sinh Π sinh sinhπ sinh sinh 3 Π 3 Π cosh sinh cosh sinh Π sinh sinh Π sinh

4 4 cosh Π cosh cosh 3 Π cosh Π cosh cosh Π cosh Π cosh 3 Π sinh cosh sinh cosh Π cosh Π sinh cosh sinh cosh Π sinh Π cosh sinh cosh Π cosh coshπ cosh cosh 3 Π 3 Π sinh cosh sinh cosh Π cosh cosh Π cosh tanh Π tanh tanh Π tanh Π tanh Π coth tanh coth tanh Π coth Π tanh coth tanh Π tanh tanhπ tanh coth Π coth coth Π coth Π coth Π tanh coth tanh coth Π tanh Π coth tanh coth Π coth cothπ coth csch Π csch csch 3 Π csch Π csch csch Π csch Π csch 3 Π sech csch sech csch Π csch Π sech csch sech csch Π sech Π csch sech csch Π csch cschπ csch csch 3 Π 3 Π sech csch sech csch Π csch csch Π csch sech Π sech sech 3 Π sech Π sech sech Π sech Π sech 3 Π csch sech csch sech Π sech Π csch sech csch sech Π csch Π sech csch sech Π sech sechπ sech sech 3 Π 3 Π csch sech csch sech Π sech sech Π sech. The best-known properties and formulas for hyperbolic functions Real values for real arguments For real values of argument, the values of all the hyperbolic functions are real (or infinity).

5 5 In the points Π n m ; n m, the values of the hyperbolic functions are algebraic. In several cases, they can even be rational numbers,, or (e.g. sinhπ, sech0, or coshπ 3 ). They can be expressed using only square roots if n and m is a product of a power of and distinct Fermat primes {3, 5, 7, 57, }. Simple values at ero All hyperbolic functions has rather simple values for arguments 0 and Π : sinh0 0 cosh0 tanh0 0 coth0 csch0 sech0 sinh Π cosh Π 0 tanh Π coth Π 0 csch Π sech Π. Analyticity All hyperbolic functions are defined for all complex values of, and they are analytical functions of over the whole complex -plane and do not have branch cuts or branch points. The two functions sinh and cosh are entire functions with an essential singular point at. All other hyperbolic functions are meromorphic functions with simple poles at points Π k ; k (for csch and coth) and at points Π Π k ; k (for sech and tanh). Periodicity All hyperbolic functions are periodic functions with a real period (Π or Π ): sinh sinh Π cosh cosh Π tanh tanh Π coth coth Π csch csch Π sech sech Π sinh Π k sinh ; k cosh Π k cosh ; k tanh Π k tanh ; k coth Π k coth ; k csch Π k csch ; k sech Π k sech ; k. Parity and symmetry All hyperbolic functions have parity (either odd or even) and mirror symmetry: sinh sinh cosh cosh tanh tanh coth coth csch csch sech sech sinh sinh cosh cosh tanh tanh coth coth csch csch sech sech. Simple representations of derivatives

6 6 The derivatives of all hyperbolic functions have simple representations that can be expressed through other hyperbolic functions: sinh coth cosh csch cosh csch sinh coth csch tanh sech sech sech tanh. Simple differential equations The solutions of the simplest second-order linear ordinary differential equation with constant coefficients can be represented through sinh and cosh. The other hyperbolic functions satisfy first-order nonlinear differential equations: w w 0 ; w cosh w0 w 0 0 w w 0 ; w sinh w0 0 w 0 w w 0 ; w c cosh c sinh. All six hyperbolic functions satisfy first-order nonlinear differential equations: w w 0 ; w sinh w0 0 Im Π w w 0 ; w cosh w0 Im Π w w 0 ; w tanh w0 0 w w 0 ; w coth w Π 0 w w 4 w 0 ; w csch w w 4 w 0 ; w sech. Applications of hyperbolic functions Trigonometric functions are intimately related to triangle geometry. Functions like sine and cosine are often introduced as edge lengths of right-angled triangles. Hyperbolic functions occur in the theory of triangles in hyperbolic spaces. Lobachevsky (89) and J. Bolyai (83) independently recognied that Euclid's fifth postulate saying that for a given line and a point not on the line, there is exactly one line parallel to the first might be changed and still be a consistent geometry. In the hyperbolic geometry it is allowable for more than one line to be parallel to the first (meaning that the parallel lines will never meet the first, however far they are extended). Translated into triangles, this means that the sum of the three angles is always less than Π. A particularly nice representation of the hyperbolic geometry can be realied in the unit disk of complex numbers (the Poincaré disk model). In this model, points are complex numbers in the unit disk, and the lines are either arcs of circles that meet the boundary of the unit circle orthogonal or diameters of the unit circle. The distance d between two points (meaning complex numbers) A and B in the Poincaré disk is: da, B tanh A B BA.

7 7 The 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: cosα AB AC A B A C. AB AC 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 denoted through 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: sinhbsinhccosα coshbcoshc cosha sinhasinhccosβ coshacoshc coshb sinhasinhbcosγ coshacoshb coshc sinβsinγcosha cosβcosγ cosα sinαsinγcoshb cosαcosα cosβ sinαsinβcoshc cosαcosβ cosγ. The sine rule for hyperbolic triangles is: sinα sinha sinβ sinγ sinhb sinhc. For a right-angle triangle, the hyperbolic version of the Pythagorean theorem follows from the preceding formulas (the right angle is taken at vertex A): cosha coshbcoshc. Using the series expansion coshx x at small scales the hyperbolic geometry is approximated by the familar Euclidean geometry. The cosine formulas and the sine formulas for hyperbolic triangles with a right angle at vertex A become: cosβ tanhc sinhb, sinβ tanha sinha cosγ tanhb sinhhc, sinγ tanha tanha. The inscribed circle has the radius: Ρ tanh cos Α cos Β cos Γ cosαcosβcosγ cosα cosβ cosγ. The circumscribed circle has the radius: Ρ tanh 4 sinh a sinh b sinh c sinγsinhasinhb.

8 f 8 Other applications As 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 Sine Function Defining the hyperbolic sine function The hyperbolic sine function is an old mathematical function. It was first used in the works of V. Riccati (757), D. Foncenex (759), and J. H. Lambert (768). The hyperbolic sine function is easily defined as the half difference of two exponential functions in the points and : sinh. After comparison with the famous Euler formula for sine (sin ), it is easy to derive the following representation of the hyperbolic sine through the circular sine: sinh sin. This formula allows the derivation of all the properties and formulas for the hyperbolic sine from the corresponding properties and formulas for the circular sine. The following formula can sometimes be used as an equivalent definition of the hyperbolic sine function: sinh k 0 k k. This series converges for all finite numbers. A quick look at the hyperbolic sine function Here is a graphic of the hyperbolic sine function f x sinhx for real values of its argument x x

9 9 Representation through more general functions The function sinh is a particular case of more complicated mathematical functions. For example, it is a special case of the generalied hypergeometric function 0 F ; a; w with the parameter a 3 at w, multiplied by : 4 sinh 0 F ; 3 ; 4. It is also a particular case of the modified Bessel function I Ν with the parameter Ν Π, multiplied by : sinh Π I. Other Bessel functions can also be expressed through the hyperbolic sine function for similar values of the parameter: sinh Π J sinh Π Y sinh Π K K. Struve functions can also degenerate into the hyperbolic sine function for a similar parameter value: sinh Π H sinh Π L. But the function sinh is also a degenerate case of the doubly periodic Jacobi elliptic functions when their second parameter is equal to 0 or : sinh sd 0 sn 0 sinh sc sd. Finally, the function sinh is the particular case of another class of functions the Mathieu functions: sinh Se, 0,. Definition of the hyperbolic sine function for a complex argument In the complex -plane, the function sinh is defined by the same formula used for real values: sinh. Here are two graphics showing the real and imaginary parts of the hyperbolic sine function over the complex plane.

10 Re y 0 5 Im y 0 x 0 x 5 The best-known properties and formulas for the hyperbolic sine function Values in points The values of the hyperbolic sine function for special values of its argument can be easily derived from the corresponding values of the circular sine in special points of the circle: sinh0 0 sinh Π 6 sinh Π 4 sinh Π 3 3 sinh Π Π 3 sinh 3 sinh 3 Π 4 sinh 5 Π 6 sinhπ 0 sinh 7 Π 6 sinh 5 Π 4 sinh 4 Π 3 3 sinh 3 Π 5 Π 3 sinh 3 sinh 7 Π 4 sinh Π 6 sinh Π 0 sinhπ m 0 ; m sinhπ m m ; m. The values at infinity can be expressed by the following formulas: sinh sinh. General characteristics For real values of argument, the values of sinh are real. In the points Π n m ; n m, the values of sinh are algebraic. In several cases, they can even be rational numbers, 0, or, multiplied by. Here are some examples: sinh0 0 sinh Π 6 sinh Π. The values of sinh n Π can be expressed using only square roots if n and m is a product of a power of and m distinct Fermat primes {3, 5, 7, 57, }.

11 The function sinh is an entire analytical function of that is defined over the whole complex -plane and does not have branch cuts and branch points. It has an essential singular point at. It is a periodic function with the period Π : sinh Π sinh sinh sinh Π k ; k sinh k sinh Π k ; k. The function sinh is an odd function with mirror symmetry: sinh sinh sinh sinh. Differentiation The derivatives of sinh have simple representations using either the sinh function or the cosh function: sinh cosh n sinh n n sinh Π n ; n. Ordinary differential equation The function sinh satisfies the simplest possible linear differential equation with constant coefficients: w w 0 ; w sinh w0 0 w 0. The complete solution of this equation can be represented as a linear combination of sinh and cosh with arbitrary constant coefficients c and c : w w 0 ; w c cosh c sinh. The function sinh also satisfies the first-order nonlinear differential equation: w w 0 ; w sinh w0 0 Im Π. Series representation The function sinh has a simple series expansion at the origin that converges in the whole complex -plane: sinh 3 k 3 k 0 k. Product representation The following infinite product representation for sinh clearly illustrates that sinh 0 at Π k k : sinh k. Π k Indefinite integration

12 Indefinite integrals of expressions involving the hyperbolic sine function can sometimes be expressed using elementary functions. However, special functions are frequently needed to express the results even when the integrands have a simple form (if they can be evaluated in closed form). Here are some examples: sinh cosh sinh sinh sinh E Π Α sinh v a v v v mod Α Α v v v v v Α k v k k 0 a n k Α Π v Α, a n k a k n Α Π v a n k Α Α, a k n ; Α 0 v. The last integral cannot be evaluated in closed form using the known classical special functions for arbitrary values of parameters Α and v. Definite integration Definite integrals that contain the hyperbolic sine are sometimes simple as shown in the following example: log3 t 0 sinht. Some special functions can be used to evaluate more complicated definite integrals. For example, gamma and polygamma functions, are needed to express the following integrals: 0 sinh a tt a a Π ; Rea 0 0 t sinh a tt 3 Rea 0. Integral transforms 8 Π a a Ψ a Ψ a Ψ a Ψ a Ψ a Ψ a ; Numerous formulas for integral transforms from circular sine functions cannot be easily converted into corresponding formulas with the hyperbolic sine function because the hyperbolic sine grows exponentially at infinity. This holds for the Fourier cosine and sine transforms, and for Mellin, Hilbert, Hankel, and other transforms. The exceptional case is the Laplace transform that itself includes the exponential function in the kernel: t sinht. Finite summation The following finite sum of the hyperbolic sine can be expressed using the hyperbolic sine function:

13 3 n sinhx k csch x x n sinh k 0 sinh x n. Infinite summation The following infinite sum can be expressed using elementary functions: sinhk x k coshx sinhsinhx. k Finite products The following finite products of the hyperbolic sine can be expressed using elementary functions and formulas: n sinh Π k n k n n sinhn sinh ; n n sinh Π k n n n k Infinite products cos n Π n Π n cos csch ; n. The following infinite product can be expressed using the hyperbolic sine function: k 4 3 sinh sinh 3 k. Addition formulas The hyperbolic sine of a sum can be represented by the rule: "the hyperbolic sine of a sum is equal to the product of the hyperbolic sine by the hyperbolic cosine plus the hyperbolic cosine by the hyperbolic sine." A similar rule is valid for the hyperbolic sine of the difference: sinha b coshb sinha cosha sinhb sinha b coshb sinha cosha sinhb. Multiple arguments In the case of multiple arguments, 3,, the function sinhn can be represented as the finite sum of terms that include powers of the hyperbolic sine and cosine: sinh sinh cosh sinh3 4 sinh 3 3 sinh sinhn n n k 0 k sinh k cosh nk ; n. The function sinhn can also be represented as the finite product including the hyperbolic sine of the linear argument of :

14 4 n sinhn n n sinh Π k n k 0 ; n. Half-angle formulas The hyperbolic sine of the half-angle can be represented by the following simple formula that is valid in half of the horiontal strip: sinh cosh ; Re 0 Π Im Π Re 0 Π Im 3 Π Re 0 0 Im Π Im Π. To make this formula correct for all complex, a complicated prefactor is needed: sinh c cosh ; c Im Im Π Im Π Π Π ΘRe ;, Π where c contains the unit step, real part, imaginary part, and the floor functions. Sums of two direct functions The sum of two hyperbolic sine functions can be described by the rule: "the sum of hyperbolic sines is equal to the doubled hyperbolic cosine of the half-difference multiplied by the hyperbolic sine of the half-sum". A similar rule is valid for the difference of two hyperbolic sines: sinha sinhb cosh a b sinha sinhb cosh a b sinh a b sinh a b. Products involving the direct function The product of two hyperbolic sine functions or the product of the hyperbolic sine and hyperbolic cosine have the following representations: sinha sinhb cosha b cosha b sinha coshb sinha b sinha b. Powers of the direct function The integer powers of the hyperbolic sine functions can be expanded as finite sums of hyperbolic cosine (or sine) functions with multiple arguments. These sums include binomial coefficients: sinh n n n n n n n k k 0 n k cosh n k ; n sinh n n n k 0 k n k sinh n k ; n.

15 5 These formulas can be combined into the following formula: sinh n n n n n n n mod n n k n k k 0 Π n cosh n k ; n. Inequalities The most famous inequality for the hyperbolic sine function can be described by the following formula: sinh sin. Relations with its inverse function There is a simple relation between the function sinh and its inverse function sinh : sinhsinh sinh sinh ; Π Im Π Im Π Re 0 Im Π Re 0. The second formula is valid at least in the horiontal strip Π Re Π. Outside this strip, a much more complicated relation (that contains the unit step, real part, imaginary part, and the floor functions) holds: sinh Im sinh Π Im Π Im Π ΘRe Π Im Π Π Im Π Im Π ΘRe. Representations through other hyperbolic functions Hyperbolic sine and cosine functions are connected by a very simple formula that contains the linear function in the argument: sinh cosh Π. Another famous formula, connecting sinh and cosh, is expressed in the analog of the well-known Pythagorean theorem: sinh cosh sinh cosh ; Re 0 3 Π Im mod Π Π Re 0 0 Im mod Π Π Re 0 Π 3 Π Im mod Π Re 0 0 Im mod Π Π Im mod Π Π. The last restriction on can be removed, but the formula will get a complicated coefficient c that contains the unit step, real part, imaginary part, and the floor function and c() : sinh c cosh ; c Im Π Im Im Π Π Π ΘRe ;. Π The hyperbolic sine function can also be represented using other hyperbolic functions by the following formulas:

16 6 sinh tanh tanh sinh coth coth sinh csch sinh. sech Π Representations through hyperbolic functions The hyperbolic sine function has representations using the other hyperbolic functions: sinh sin sinh sin sinh cos Π sinh tan tan sinh cot csc cot sinh sinh. sec Π Applications The hyperbolic sine function is used throughout mathematics, the exact sciences, and engineering. Introduction to the Hyperbolic Functions in Mathematica Overview The following shows how the six hyperbolic functions are realied in Mathematica. Examples of evaluating Mathematica functions applied to various numeric and exact expressions that involve the hyperbolic functions or return them are shown. These involve numeric and symbolic calculations and plots. Notations Mathematica forms of notations All six hyperbolic functions are represented as built-in functions in Mathematica. Following Mathematica's general naming convention, the StandardForm function names are simply capitalied versions of the traditional mathematics names. Here is a list hypfunctions of the six hyperbolic functions in StandardForm. hypfunctions Sinh, Cosh, Tanh, Coth, Sech, Cosh Sinh, Cosh, Tanh, Coth, Sech, Cosh Here is a list hypfunctions of the six trigonometric functions in TraditionalForm. hypfunctions TraditionalForm sinh, cosh, tanh, coth, sech, cosh Additional forms of notations Mathematica also knows the most popular forms of notations for the hyperbolic functions that are used in other programming languages. Here are three examples: CForm, TeXForm, and FortranForm. hypfunctions. Π CForm

17 7 List(Sinh(*Pi*),Cosh(*Pi*),Tanh(*Pi*),Coth(*Pi*),Sech(*Pi*),Cosh(*Pi*)) hypfunctions. Π TeXForm \{ \sinh (\,\pi \,),\cosh (\,\pi \,),\tanh (\,\pi \,),\coth (\,\pi \,), \Mfunction{Sech}(\,\pi \,),\cosh (\,\pi \,)\} hypfunctions. Π FortranForm List(Sinh(*Pi*),Cosh(*Pi*),Tanh(*Pi*),Coth(*Pi*),Sech(*Pi*),Cosh(*Pi*)) Automatic evaluations and transformations Evaluation for exact, machine-number, and high-precision arguments For a simple exact argument, Mathematica returns an exact result. For instance, for the argument Π 6, the Sinh function evaluates to. Sinh Π 6 Sinh, Cosh, Tanh, Coth, Csch, Sech. Π 6, 3, 3, 3,, 3 For a generic machine-number argument (a numerical argument with a decimal point and not too many digits), a machine number is returned. Cosh Sinh, Cosh, Tanh, Coth, Csch, Sech , 3.76, ,.0373, 0.757, The next inputs calculate 00-digit approximations of the six hyperbolic functions at. NTanh, Coth N, 50 & NSinh, Cosh, Tanh, Coth, Csch, Sech., 00

18 , , , , , Within a second, it is possible to calculate thousands of digits for the hyperbolic functions. The next input calculates 0000 digits for sinh, cosh, tanh, coth, sech, and csch and analyes the frequency of the occurrence of the digit k in the resulting decimal number. MapFunctionw, First, Length & SplitSortFirstRealDigitsw, NSinh, Cosh, Tanh, Coth, Csch, Sech., , 980,, 994,, 996, 3, 04, 4, 986, 5, 00, 6, 07, 7, 00, 8, 98, 9, 0, 0, 05,, 960,, 997, 3, 037, 4, 070, 5, 08, 6, 973, 7, 997, 8, 963, 9, 970, 0, 97,, 03,, 06, 3, 970, 4, 949, 5, 05, 6, 98, 7, 056, 8, 00, 9, 97, 0, 975,, 986,, 03, 3, 004, 4, 008, 5, 977, 6, 977, 7, 036, 8, 035, 9, 979, 0, 979,, 030,, 987, 3, 99, 4, 06, 5, 030, 6, 0, 7, 969, 8, 974, 9, 00, 0, 009,, 97,, 08, 3, 994, 4, 0, 5, 08, 6, 958, 7, 09, 8, 06, 9, 986 Here are 50-digit approximations to the six hyperbolic functions at the complex argument 3 5. NCsch3 5, NSinh, Cosh, Tanh, Coth, Csch, Sech. 3 5, 50

19 , , , , , Mathematica always evaluates mathematical functions with machine precision, if the arguments are machine numbers. In this case, only six digits after the decimal point are shown in the results. The remaining digits are suppressed, but can be displayed using the function InputForm. Sinh., NSinh, NSinh, 6, NSinh, 5, NSinh, , , , , InputForm { , , , , `0} Precision 6 Simplification of the argument Mathematica uses symmetries and periodicities of all the hyperbolic functions to simplify expressions. Here are some examples. Sinh Sinh Sinh Π Sinh Sinh Π Sinh Sinh 34Π Sinh Sinh, Cosh, Tanh, Coth, Csch, Sech Sinh, Cosh, Tanh, Coth, Csch, Sech

20 0 Sinh Π, Cosh Π, Tanh Π, Coth Π, Csch Π, Sech Π Sinh, Cosh, Tanh, Coth, Csch, Sech Sinh Π, Cosh Π, Tanh Π, Coth Π, Csch Π, Sech Π Sinh, Cosh, Tanh, Coth, Csch, Sech Sinh 34Π, Cosh 34Π, Tanh 34Π, Coth 34Π, Csch 34Π, Sech 34Π Sinh, Cosh, Tanh, Coth, Csch, Sech Mathematica automatically simplifies the composition of the direct and the inverse hyperbolic functions into the argument. SinhArcSinh, CoshArcCosh, TanhArcTanh, CothArcCoth, CschArcCsch, SechArcSech,,,,, Mathematica also automatically simplifies the composition of the direct and any of the inverse hyperbolic functions into algebraic functions of the argument. SinhArcSinh, SinhArcCosh, SinhArcTanh, SinhArcCoth, SinhArcCsch, SinhArcSech,,,,, CoshArcSinh, CoshArcCosh, CoshArcTanh, CoshArcCoth, CoshArcCsch, CoshArcSech,,,,, TanhArcSinh, TanhArcCosh, TanhArcTanh, TanhArcCoth, TanhArcCsch, TanhArcSech,,,,, CothArcSinh, CothArcCosh, CothArcTanh, CothArcCoth, CothArcCsch, CothArcSech

21 CschArcSinh, CschArcCosh, CschArcTanh, CschArcCoth, CschArcCsch, CschArcSech,,,,, SechArcSinh, SechArcCosh, SechArcTanh, SechArcCoth, SechArcCsch, SechArcSech,,,,, In cases where the argument has the structure Π k or Π k, e Π k or Π k with integer k, trigonometric functions can be automatically transformed into other trigonometric or hyperbolic functions. Here are some examples. Tanh Π Coth Csch Csc Sinh Π Π Π Π Π Π, Cosh, Tanh, Coth, Csch, Sech Cosh, Sinh, Coth, Tanh, Sech, Csch Sinh, Cosh, Tanh, Coth, Csch, Sech Sin, Cos, Tan, Cot, Csc, Sec Simplification of simple expressions containing hyperbolic functions Sometimes simple arithmetic operations containing hyperbolic functions can automatically produce other hyperbolic functions. Sech Cosh Sinh, Cosh, Tanh, Coth, Csch, Sech, Sinh Cosh, Cosh Sinh, Sinh SinhΠ, Cosh Sinh^

22 Csch, Sech, Coth, Tanh, Sinh, Cosh, Tanh, Coth, Tanh, Coth Csch Hyperbolic functions as special cases of more general functions All hyperbolic functions can be treated as particular cases of some more advanced special functions. For example, sinh and cosh are sometimes the results of auto-simplifications from Bessel, Mathieu, Jacobi, hypergeometric, and Meijer functions (for appropriate values of their parameters). BesselI, Π Sinh MathieuC, 0, Cosh JacobiSN, Tanh BesselI,, MathieuS, 0,, JacobiSD, 0, HypergeometricPFQ, 3, 4, MeijerG,, 0,, 4 Π Sinh, Sinh, Sinh, Sinh, Sinh Π BesselI,, MathieuC, 0,, JacobiCD, 0, Hypergeometric0F, 4, MeijerG,, 0,, 4 Π Cosh, Cosh, Cosh, Cosh, Cosh Π JacobiSC, 0, JacobiNS,, JacobiNS, 0, JacobiDC, 0 Tanh, Coth, Csch, Sech Equivalence transformations carried out by specialied Mathematica functions General remarks

23 3 Automatic evaluation and transformations can sometimes be inconvenient: They act in only one chosen direction and the result can be overly complicated. For example, the expression cosh is generally preferable to the more complicated sinhπ coshπ 3. Mathematica provides automatic transformation of the second expression into the first one. But compact expressions like sinhcoshπ 6 should not be automatically expanded into the more complicated expression sinhcosh. Mathematica has special functions that produce these types of expansions. Some of them are demonstrated in the next section. TrigExpand The function TrigExpand expands out trigonometric and hyperbolic functions. In more detail, it splits up sums and integer multiples that appear in the arguments of trigonometric and hyperbolic functions, and then expands out the products of the trigonometric and hyperbolic functions into sums of powers, using the trigonometric and hyperbolic identities where possible. Here are some examples. TrigExpandSinhx y Coshy Sinhx Coshx Sinhy Cosh4 TrigExpand Cosh 4 6 Cosh Sinh Sinh 4 TrigExpandSinhx y, Sinh3, Coshx y, Cosh3, Tanhx y, Tanh3, Cothx y, Coth3, Cschx y, Csch3, Sechx y, Sech3 Coshy Sinhx Coshx Sinhy, 3 Cosh Sinh Sinh 3, Coshx Coshy Sinhx Sinhy, Cosh 3 3 Cosh Sinh, Coshy Sinhx Coshx Coshy Sinhx Sinhy Coshx Sinhy Coshx Coshy Sinhx Sinhy, 3 Cosh Sinh Cosh 3 3 Cosh Sinh Sinh 3 Cosh 3 3 Cosh Sinh, Coshx Coshy Coshy Sinhx Coshx Sinhy Sinhx Sinhy Coshy Sinhx Coshx Sinhy, Cosh 3 3 Cosh Sinh, 3 Cosh Sinh Sinh 3 3 Cosh Sinh Sinh 3 Coshy Sinhx Coshx Sinhy,, 3 Cosh Sinh Sinh 3 Coshx Coshy Sinhx Sinhy, Cosh 3 3 Cosh Sinh TableForm TrigExpand & FlattenSinhx y, Sinh3, Coshx y, Cosh3, Tanhx y, Tanh3, Cothx y, Coth3, Cschx y, Csch3, Sechx y, Sech3

24 4 Sinhx y Coshy Sinhx Coshx Sinhy Sinh3 3 Cosh Sinh Sinh 3 Coshx y Coshx Coshy Sinhx Sinhy Cosh3 Cosh 3 3 Cosh Sinh Tanhx y Tanh3 Cothx y Coth3 Cschx y Csch3 Sechx y Sech3 TrigFactor Coshy Sinhx Coshx CoshySinhx Sinhy Coshx Sinhy Coshx CoshySinhx Sinhy 3 Cosh Sinh 3 Sinh Cosh 3 3 Cosh Sinh Cosh 3 3 Cosh Sinh Coshx Coshy Coshy SinhxCoshx Sinhy Sinhx Sinhy Coshy SinhxCoshx Sinhy Cosh 3 3 Cosh Sinh 3 Cosh SinhSinh 3 3 Cosh SinhSinh 3 Coshy SinhxCoshx Sinhy 3 Cosh SinhSinh 3 Coshx CoshySinhx Sinhy Cosh 3 3 Cosh Sinh The command TrigFactor factors trigonometric and hyperbolic functions. In more detail, it splits up sums and integer multiples that appear in the arguments of trigonometric and hyperbolic functions, and then factors the resulting polynomials in the trigonometric and hyperbolic functions, using the corresponding identities where possible. Here are some examples. TrigFactorSinhx Coshy Cosh x y Sinh x y Cosh x y Sinh x y Tanhx Cothy TrigFactor Coshx y Cschy Sechx TrigFactorSinhx Sinhy, Coshx Coshy, Tanhx Tanhy, Cothx Cothy, Cschx Cschy, Sechx Sechy Cosh x y Sinh x y, Cosh x y Cosh x y, Sechx Sechy Sinhx y, Cschx Cschy Sinhx y, Cosh x y Csch x Csch y Sech x Sech y Sinh x y, Cosh x y Cosh x y Cosh x Sinh x Cosh x Sinh x Cosh y Sinh y Cosh y Sinh y TrigReduce

25 5 The command TrigReduce rewrites products and powers of trigonometric and hyperbolic functions in terms of those functions with combined arguments. In more detail, it typically yields a linear expression involving trigonometric and hyperbolic functions with more complicated arguments. TrigReduce is approximately opposite to TrigExpand and TrigFactor. Here are some examples. TrigReduceSinh^3 3 Sinh Sinh3 4 SinhxCoshy TrigReduce Sinhx y Sinhx y TrigReduceSinh^, Cosh^, Tanh^, Coth^, Csch^, Sech^ Cosh, Cosh, Cosh Cosh, Cosh Cosh, Cosh, Cosh TrigReduceTrigExpandSinhx y, Sinh3, SinhxSinhy, Coshx y, Cosh3, CoshxCoshy, Tanhx y, Tanh3, TanhxTanhy, Cothx y, Coth3, CothxCothy, Cschx y, Csch3, CschxCschy, Sechx y, Sech3, SechxSechy Sinhx y, Sinh3, Coshx y Coshx y, Coshx y, Cosh3, Coshx y Coshx y, Coshx y Coshx y Tanhx y, Tanh3, Coshx y Coshx y, Cothx y, Coth3, Coshx y Coshx y Coshx y Coshx y, Cschx y, Csch3, Coshx y Coshx y, Sechx y, Sech3, Coshx y Coshx y TrigReduceTrigFactorSinhx Sinhy, Coshx Coshy, Tanhx Tanhy, Cothx Cothy, Cschx Cschy, Sechx Sechy Sinhx Sinhy, Coshx Coshy, Sinhx y Coshx y Coshx y, Sinhx y Coshx y Coshx y, Sinhx Sinhy Coshx y Coshx y, TrigToExp Coshx Coshy Coshx y Coshx y

26 6 The command TrigToExp converts direct and inverse trigonometric and hyperbolic functions to exponentials or logarithmic functions. It tries, where possible, to give results that do not involve explicit complex numbers. Here are some examples. TrigToExpSinh SinhTanh TrigToExp TrigToExpSinh, Cosh, Tanh, Coth, Csch, Sech,,,,, ExpToTrig The command ExpToTrig converts exponentials to trigonometric or hyperbolic functions. It tries, where possible, to give results that do not involve explicit complex numbers. It is approximately opposite to TrigToExp. Here are some examples. ExpToTrig x Β Coshx Β Sinhx Β x Α x Β ExpToTrig x Γ x Coshx Α Coshx Β Sinhx Α Sinhx Β Coshx Γ Coshx Sinhx Γ Sinhx ExpToTrigTrigToExpSinh, Cosh, Tanh, Coth, Csch, Sech Sinh, Cosh, Tanh, Coth, Csch, Sech ExpToTrigΑ x Β Α x Β, Α x Β Γ x Β Α Coshx Β, Γ Cosx Β Α Coshx Β Γ Sinx Β Α Sinhx Β ComplexExpand The function ComplexExpand expands expressions assuming that all the occurring variables are real. The value option TargetFunctions is a list of functions from the set {Re, Im, Abs, Arg, Conjugate, Sign}. ComplexExpand tries to give results in terms of the specified functions. Here are some examples. ComplexExpandSinhx ycoshx y Cosy Coshx Sinhx Coshx Siny Sinhx Cosy Coshx Siny Cosy Siny Sinhx

27 7 Cschx ysechx y ComplexExpand 4 Cosy Coshx Sinhx Cos y Cosh x Cos y Cosh x 4 Coshx Siny Sinhx Cos y Cosh x Cos y Cosh x 4 Cosy Coshx Siny Cos y Cosh x Cos y Cosh x 4 Cosy Siny Sinhx Cos y Cosh x Cos y Cosh x li Sinhx y, Coshx y, Tanhx y, Cothx y, Cschx y, Sechx y Sinhx y, Coshx y, Tanhx y, Cothx y, Cschx y, Sechx y ComplexExpandli Coshx Siny Cosy Sinhx, Cosy Coshx Siny Sinhx, Sin y Cos y Cosh x Coshx Siny Cos y Cosh x Sinh x Cos y Cosh x, Cosy Sinhx Cos y Cosh x, Sin y Cos y Cosh x Cosy Coshx Cos y Cosh x ComplexExpandRe & li, TargetFunctions Re, Im Cosy Sinhx, Cosy Coshx, Sinh x Cos y Cosh x, Sinh x Cos y Cosh x, Cosy Sinhx Cos y Cosh x, Cosy Coshx Cos y Cosh x ComplexExpandIm & li, TargetFunctions Re, Im Coshx Siny, Siny Sinhx, Sin y Cos y Cosh x, Sin y Cos y Cosh x, Coshx Siny Cos y Cosh x, Siny Sinhx Cos y Cosh x ComplexExpandAbs & li, TargetFunctions Re, Im Sinh x Cos y Cosh x, Siny Sinhx Cos y Cosh x

28 8 Coshx Siny Cosy Sinhx, Cosy Coshx Siny Sinhx, Sin y Sinh x Cos y Cosh x Cos y Cosh x, Sin y Sinh x Cos y Cosh x Cos y Cosh x, 4 Coshx Siny 4 Cosy Sinhx Cos y Cosh x Cos y Cosh x, 4 Cosy Coshx 4 Siny Sinhx Cos y Cosh x Cos y Cosh x Simplify, x, y Reals & Cos y Cosh x, Cos y Cosh x, Sin y Sinh x Cos y Cosh x, Cos y Cosh x Cos y Cosh x, Cos y Cosh x, Cos y Cosh x ComplexExpandArg & li, TargetFunctions Re, Im ArcTanCosy Sinhx, Coshx Siny, ArcTanCosy Coshx, Siny Sinhx, Sinh x ArcTan Cos y Cosh x, Sinh x ArcTan Cos y Cosh x, Cosy Sinhx ArcTan Cos y Cosh x, Sin y Cos y Cosh x, Sin y Cos y Cosh x, Coshx Siny Cos y Cosh x, Cosy Coshx ArcTan Cos y Cosh x, Siny Sinhx Cos y Cosh x Simplify, x, y Reals & ArcTanCosy Sinhx, Coshx Siny, ArcTanCosy Coshx, Siny Sinhx, ArcTanSinh x, Sin y, ArcTanCoshx Sinhx, Cosy Siny, ArcTanCosy Sinhx, Coshx Siny, ArcTanCosy Coshx, Siny Sinhx ComplexExpandConjugate & li, TargetFunctions Re, Im Simplify

29 9 Coshx Siny Cosy Sinhx, Cosy Coshx Siny Sinhx, Sin y Sinh x Sin y Sinh x, Cos y Cosh x Cos y Cosh x, Coshx Siny Cosy Sinhx, Cosy Coshx Siny Sinhx Simplify The command Simplify performs a sequence of algebraic transformations on an expression, and returns the simplest form it finds. Here are two examples. SimplifySinh Sinh Cosh Sinh Cosh Simplify Sinh Here is a large collection of hyperbolic identities. All are written as one large logical conjunction.

30 30 Simplify & Cosh Sinh Sinh Cosh Cosh Cosh Cosh Tanh Cosh Cosh Coth Cosh Sinh SinhCosh Cosh Cosh Sinh Cosh Sinha b SinhaCoshb CoshaSinhb Sinha b SinhaCoshb CoshaSinhb Cosha b CoshaCoshb SinhaSinhb Cosha b CoshaCoshb SinhaSinhb Sinha Sinhb Sinh a b a b Cosh Sinha Sinhb Cosh a b a b Sinh Cosha Coshb Cosh a b a b Cosh Cosha Coshb Sinh a b b a Sinh Sinha b Sinha Tanha Tanhb Tanha Tanhb CoshaCoshb b CoshaCoshb A Sinh B Cosh A B A Sinh ArcTanh B A Cosha b Cosha b SinhaSinhb CoshaCoshb Cosha b Cosha b Sinha b Sinha b SinhaCoshb Sinh Cosh Cosh Cosh Tanh Cosh Sinh Sinh Cosh Coth Sinh Cosh Cosh Sinh True The command Simplify has the Assumption option. For example, Mathematica knows that sinhx 0 for all real positive x, and uses the periodicity of hyperbolic functions for the symbolic integer coefficient k of k Π. SimplifyAbsSinhx 0, x 0 True AbsSinhx 0 Simplify, x 0 &

31 3 True SimplifySinh k Π, Cosh k Π, Tanh k Π, Coth k Π, Csch k Π, Sech k Π, k Integers Sinh, Cosh, Tanh, Coth, Csch, Sech SimplifySinh k Π Sinh, Cosh k Π Cosh, Tanh k Π Tanh, Coth k Π Coth, Csch k Π Csch, Sech k Π Sech, k Integers k, k,,, k, k Mathematica also knows that the composition of inverse and direct hyperbolic functions produces the value of the inner argument under the appropriate restriction. Here are some examples. SimplifyArcSinhSinh, ArcTanhTanh, ArcCothCoth, ArcCschCsch, Π Im Π,,, SimplifyArcCoshCosh, ArcSechSech, Π Im Π Re 0, FunctionExpand (and Together) While the hyperbolic functions auto-evaluate for simple fractions of Π, for more complicated cases they stay as hyperbolic functions to avoid the build up of large expressions. Using the function FunctionExpand, such expressions can be transformed into explicit radicals. Cosh Π 3 Cos Π 3 FunctionExpand Cosh Π 3 Coth Π FunctionExpand

32 3 Sinh Π Π Π Π Π Π, Cosh, Tanh, Coth, Csch, Sech Sin Π 6, Cos Π 6, Tan Π 6, Cot Π 6, Csc Π 6, Sec Π 6 FunctionExpand[%],,,,, Sinh Π Π Π Π Π Π, Cosh, Tanh, Coth, Csch, Sech Sin Π 60, Cos Π 60, Tan Π 60, Cot Π 60, Csc Π 60, Sec Π 60 TogetherFunctionExpand , , , , ,

33 33 If the denominator contains squares of integers other than, the results always contain complex numbers (meaning that the imaginary number appears unavoidably). Sinh Π Π Π Π Π Π, Cosh, Tanh, Coth, Csch, Sech Sin Π 9, Cos Π 9, Tan Π 9, Cot Π 9, Csc Π 9, Sec Π 9 FunctionExpand[%] // Together , , , , , Here the function RootReduce is used to express the previous algebraic numbers as numbered roots of polynomial equations. RootReduceSimplify Root &, 4, Root &, 3, Root &, 4, Root &, 3, Root &, 5, Root8 6 3 &, 3 The function FunctionExpand also reduces hyperbolic expressions with compound arguments or compositions, including hyperbolic functions, to simpler forms. Here are some examples. FunctionExpandCoth

34 34 Cot Tanh FunctionExpand Tanh 4 Sinh, Cosh, Tanh, Coth, Csch, Sech FunctionExpand Sinh Tanh, Cosh,, Coth Csch,, Sech Applying Simplify to the last expression gives a more compact result. Simplify Sinh, Cosh, Tanh, Coth, Csch, Sech Here are some similar examples. SinhArcTanh FunctionExpand Cosh ArcCoth FunctionExpand SinhArcSinh, CoshArcCosh, TanhArcTanh, CothArcCoth, CschArcCsch, SechArcSech FunctionExpand,,,, 3,

35 35 Sinh ArcSinh, Cosh ArcCosh, Tanh ArcTanh, Coth ArcCoth, Csch ArcCsch, Sech ArcSech FunctionExpand,,,,, Simplify,,,,, FullSimplify The function FullSimplify tries a wider range of transformations than the function Simplify and returns the simplest form it finds. Here are some examples that contrast the results of applying these functions to the same expressions. Cosh Log Log Simplify Cosh Log Log FullSimplify SinhLog, CoshLog, TanhLog, CothLog, CschLog, SechLog Simplify

36 36,,,,, SinhLog, CoshLog, TanhLog, CothLog, CschLog, SechLog FullSimplify,,,,, Operations performed by specialied Mathematica functions Series expansions Calculating the series expansion of hyperbolic functions to hundreds of terms can be done in seconds. Here are some examples. SeriesSinh,, 0, O6 0 Normal SeriesSinh, Cosh, Tanh, Coth, Csch, Sech,, 0, O4, O4, 3 3 O4, 3 45 O4, O4, O4 SeriesCoth,, 0, 00 Timing

37

38 O 0

39 39 Mathematica comes with the add-on package DiscreteMath`RSolve` that allows finding the general terms of series for many functions. After loading this package, and using the package function SeriesTerm, the following n th term for odd hyperbolic functions can be evaluated. DiscreteMath`RSolve` SeriesTermSinh, Cosh, Tanh, Coth, Csch, Sech,, 0, n^n n IfOddn, n IfOddn, n, 0, n IfEvenn, n, 0, n n BernoulliB n, 0, n n n BernoulliB n, n n n BernoulliB n, n, n EulerEn n Here is a quick check of the last result. This series should be evaluated to sinh, cosh, tanh, coth, csch, sech, which can be concluded from the following relation. Sum, n, 0, 00 & SeriesSinh, Cosh, Tanh, Coth, Csch, Sech,, 0, 00 O 0, O 0, O 0, O0, O0, O 0 Differentiation Mathematica can evaluate derivatives of hyperbolic functions of an arbitrary positive integer order. DSinh, Cosh Sinh D, & Cosh Sinh, Cosh, Tanh, Coth, Csch, Sech Cosh, Sinh, Sech, Csch, Coth Csch, Sech Tanh, Sinh, Cosh, Tanh, Coth, Csch, Sech Sinh, Cosh, Sech Tanh, Coth Csch, Coth Csch Csch 3, Sech 3 Sech Tanh TableDSinh, Cosh, Tanh, Coth, Csch, Sech,, n, n, 4

40 40 Cosh, Sinh, Sech, Csch, Coth Csch, Sech Tanh, Sinh, Cosh, Sech Tanh, Coth Csch, Coth Csch Csch 3, Sech 3 Sech Tanh, Cosh, Sinh, Sech 4 4 Sech Tanh, 4 Coth Csch Csch 4, Coth 3 Csch 5 Coth Csch 3, 5 Sech 3 Tanh Sech Tanh 3, Sinh, Cosh, 6 Sech 4 Tanh 8 Sech Tanh 3, 8 Coth 3 Csch 6 Coth Csch 4, Coth 4 Csch 8 Coth Csch 3 5 Csch 5, 5 Sech 5 8 Sech 3 Tanh Sech Tanh 4 Finite summation Mathematica can calculate finite sums that contain hyperbolic functions. Here are two examples. SumSinha k, k, 0, n aa n a a n aa n a n k Sinha k k 0 a a n a a a n a Infinite summation Mathematica can calculate infinite sums that contain hyperbolic functions. Here are some examples. k Sinhk x k x x x Sinhk x k k x x Coshk x k k Log x Log x Finite products Mathematica can calculate some finite symbolic products that contain the hyperbolic functions. Here are two examples.

41 4 n Sinh Π k n k n n n Cosh Π k n k n n Sech Sin n Π Infinite products Mathematica can calculate infinite products that contain hyperbolic functions. Here are some examples. Exp k Sinhk x k x x x Coshk x Exp k k x x Indefinite integration Mathematica can calculate a huge set of doable indefinite integrals that contain hyperbolic functions. Here are some examples. Sinh7 Cosh7 7 Sinh, Sinh a, Cosh, Cosh a, Tanh, Tanh a, Coth, Coth a, Csch, Csch a, Sech, Sech a

42 4 Cosh, Cosh HypergeometricF, a, 3, Cosh Sinh a Sinh Cosh a HypergeometricF a,, 3a, Cosh Sinh Sinh,, a Sinh a, LogCosh, LogSinh, HypergeometricF a a,,, Tanh Tanh a a Coth a HypergeometricF a a,,, Coth, a, LogCosh LogSinh, Cosh Csch a HypergeometricF, a, 3, Cosh Sinh a, ArcTanTanh, HypergeometricF a,, 3a, Cosh Sech a Sinh a Sinh Definite integration Mathematica can calculate wide classes of definite integrals that contain hyperbolic functions. Here are some examples. Π 0 3 Sinh 3 Π Gamma 3 Π 3 Cosh Gamma 7 HypergeometricF, 3, 3, Cosh Π 6 Π Sinh, Cosh, Tanh, Coth, Csch, Sech

43 EllipticE 4 4 Π, 4 EllipticE Π,, 4 EllipticE Π, EllipticE 4,, Log Log Log Π Π Log Π Π Log Tanh Log Tanh Log Tanh Π Log Tanh Π, Log Log Log Π Π Log Π Log Coth Log Coth Π Log Coth Π Log Coth Π, 34 EllipticF 4 4 Π, 34 EllipticF Π,, 4 EllipticF Π, EllipticF 4, Π 4 Sinh, Sinh a, Cosh, Cosh a, Tanh, Tanh a, Coth, Coth a, Csch, Csch a, Sech, Sech a