Hyperbolic Functions
Hyperbolic functions are analogs of trigonometric functions, defined in terms of the hyperbola rather than the circle. The main hyperbolic functions are the hyperbolic sine, hyperbolic cosine, and hyperbolic tangent. They have applications in various areas of mathematics and physics, particularly in the study of curves and surfaces.
3 Key excerpts on "Hyperbolic Functions"
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...tan −1 x and cotan − 1 x ; these are plotted in Fig. 2.13 b. In both cases, argument x spans the whole real axis – but one usually restricts attention to the portion comprised between −π /2 and π /2 in the case of tan −1 x, and to the portion comprised between 0 and π in the case of cotan −1 x, so as to back up true (i.e. single‐valued) functions. 2.4 Hyperbolic Functions Exponential functions are quite useful in process engineering problems; solutions to differential equations involving an exponential of a given argument, and simultaneously of its negative are indeed frequently found. Therefore, a set of functions termed Hyperbolic Functions has been designed to assist in the associated modeling; coincidentally, they satisfy most operational relationships of trigonometric functions, and have accordingly also been termed hyperbolic trigonometric functions. 2.4.1 Definition and Major Features The two basic Hyperbolic Functions are the hyperbolic sine, sinh x, defined as (2.472) and the hyperbolic cosine, cosh x, abiding to (2.473) the plots of Eqs. (2.472) and (2.473) are provided in Fig. 2.14 a. Note the even nature of cosh x, i.e. (2.474) with the aid of Eq. (2.473) ; in contrast to the odd nature of sinh x, according to (2.475) as per Eq. (2.472). The curves representing these two functions overlap at large x, i.e. (2.476) stemming from Eqs. (2.472) and (2.473), as emphasized in Fig. 2.14 a (with the exact concept of limit coming soon); while there is a unit minimum value of cosh x at x = 0, viz. (2.477) based on Eq. (2.473) – with derivation rules to be introduced in due course. Figure 2.14 Variation, with their argument x, of major Hyperbolic Functions, (a) hyperbolic sine (sinh) and cosine (cosh), (b) hyperbolic tangent (tanh) and cotangent (cotanh), and (c) hyperbolic secant (sech) and cosecant (cosech). On the other hand, ordered addition of Eqs...
eBook - ePub- Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
- 2009(Publication Date)
- Garland Science(Publisher)
...In science, these same functions describe at least approximately the oscillations of objects such as atoms in a molecule, the propagation of sound and light waves, and the hours of daylight throughout the year. Indeed, even those cycles that are not sinusoidal are often analyzed as sums of sine and cosine functions, an advanced topic called frequency analysis, which will not be considered in this book. The properties of the trigonometric functions are easiest to understand when they are considered as functions of an angle α. Note it is conventional to use Greek letters like α, β, γ, θ, and ϕ to stand for angles. We could use any symbols we like but it is easier to communicate if we use the symbols that others expect. The definitions of sin(α) and cos(α) when α is an angle are shown in Figure 4.10. This shows a circle with radius r with a line drawn from the center to the circle at angle α relative to the x axis. We can draw a right angle triangle with base x, height y, and hypotenuse r, as shown. The trigonometric functions, angle, sine, cosine, and tangent, are defined as a n g l e : α = arc length/ r, s i n e : sin (α) = y / r, c o s i n e : cos (α) = x / r, t a n g e n t : tan (α) = sin (α) / cos (α) = y / x. (EQ4.13) Figure 4.10 Definitions of the sine, cosine, and tangent functions when the argument, α, is an angle. sin(α) = y / r, cos(α) = x / r, and tan(α) = y / x. The line, whose length is y, is perpendicular to the x axis. The definitions are very simple – the hard part is actually finding accurate values for x / r and y / r when we know the angle, α. The formulae that enable us to calculate the trigonometric functions are complicated but fortunately these have been programmed into scientific calculators or used to print tables. All we need to do is look up the values. A short table is provided in Appendix 3...
eBook - ePub- Richard C. Dorf, Ronald J. Tallarida(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
...9 Special Functions 1. Hyperbolic. Functions sinh x = e x − e − x 2 csch x = 1 sinh x cosh x = e x + e − x 2 sech x = 1 cosh x tanh x = e x − e − x e x + e − x ctnh x = 1 tanh x sinh (− x) = − sinh x ctnh (− x) = − ctnh x cosh (− x) = cosh x sech([-. -=PLGO-SEPARATOR=--]− x) = s ech x tanh (− x) = − tanh x c sch (− x) = − csch x tanh x = sin x cosh x ctnh x = cosh x sinh x cosh 2 x − sinh 2 x = 1 cosh 2 x = 1 2 (cosh 2 x + 1) sinh 2 x = 1 2 (cosh 2 x − 1) ctnh 2 x -csch 2[--=PLG. O-SEPARATOR=--]x = 1 csch 2 x − sech 2 x = csch 2 x sech 2 x tanh 2 x + sech 2 x = 1 sinh (x + y) = sinh x cosh y + cosh x sinh y cosh (x + y) = cosh x cosh y + sinh x sinh y sinh (x − y) = sinh x cosh y − cosh x sinh y cosh (x − y)[. --=PLGO-SEPARATOR=--]= cosh x cosh y − cosh x sinh y tanh (x + y) = tanh x + tanh y 1 + tanh x tanh y tanh (x − y) = tanh x − tanh y 1 − tanh x tanh y 2. Gamma Function (Generalized Factorial Function) The gamma function, denoted Γ(x), is defined. by Γ (x) = ∫ 0 ∞ e − 1 t x − 1 d t (x > 0) • Properties Γ (x + 1) = x Γ (x) (x > 0) Γ (1) = 1 Γ (n + 1) = n Γ (n) = n ! (n = 1, 2, 3, …) Γ (x) Γ (1 − x[--. =PLGO-SEPARATOR=--])) = π / sin π x Γ (1 2) = π 2 2 x − 1 Γ (x) Γ (x + 1 2) = π Γ (2 x) 3. Laplace Transforms The Laplace transform of the function f (t), denoted by F (s) or L { f (t)}, is defined F (s) = ∫ 0 ∞ f (t) e − s t d t provided that the integration may be validly performed. A sufficient condition for the existence of F (s) is that f (t) be of exponential order as t → ∞ and that it is sectionally continuous over every finite interval in the range t ≥ 0. The Laplace transform of g (t) is denoted by L { g (t)}. or G (s). • Operations f (t) F (s) = ∫ 0 ∞ f (t) e − s t d t a f (t) + b g (t) a F (s) + b G (s) f ′ (t) s F (s) − f (0) f ″ (t) s 2 F (s) − s f (0)[-. -=PLGO-SEPARATOR=--]− f ′ (0) f (n) (t) s n F (s) − s n − 1 f (0) − s n − 2 f ′ (0) − ⋯ − f (n − 1) (0) t f (t) − F ′ (s) t n f (t) (− 1) n F (n) (s) e a t f ([--=PLGO-SEPAR...
