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Hyperbolic Functions Dr. Farhana Shaheen Yanbu University College KSA Hyperbolic Functions Vincenzo Riccati (1707 - 1775) is given credit for introducing the hyperbolic functions. Hyperbolic functions are very useful in both mathematics and physics. The hyperbolic functions are: Hyperbolic sine: Hyperbolic cosine Equilateral hyperbola x = coshα , y = sinhα x2 – y2= cosh2 α - sinh2 α = 1. GRAPHS OF HYPERBOLIC FUNCTIONS y = sinh x y = cosh x Graphs of cosh and sinh functions The St. Louis arch is in the shape of a hyperbolic cosine. Hyperbolic Curves y = cosh x The curve formed by a hanging necklace is called a catenary. Its shape follows the curve of y = cosh x. Catenary Curves The curve described by a uniform, flexible chain hanging under the influence of gravity is called a catenary curve. This is the familiar curve of an electric wire hanging between two telephone poles. Catenary curves in Architecture In architecture, an inverted catenary curve is often used to create domed ceilings. This shape provides an amazing amount of structural stability as attested by fact that many of ancient structures like the pantheon of Rome which employed the catenary in their design are still standing. Masjid in Kazkhistan Fatima masjid in Kuwait Kul sharif masjid in Russia Masjid in Georgia Great Masjid in China Catenary Curve The curve is described by a COSH(theta) function Example of catenary and non-catenary curves Sinh graphs Graphs of tanh and coth functions y = tanh x y = coth x Graphs of sinh, cosh, and tanh Graphs of sech and csch functions y = sech x y = csch x Useful relations Hence: 1 - (tanh x)2 = (sech x)2. RELATIONSHIPS OF HYPERBOLIC FUNCTIONS tanh x = sinh x/cosh x coth x = 1/tanh x = cosh x/sinh x sech x = 1/cosh x csch x = 1/sinh x cosh2x - sinh2x = 1 sech2x + tanh2x = 1 coth2x - csch2x = 1 The following list shows the principal values of the inverse hyperbolic functions expressed in terms of logarithmic functions which are taken as real valued. sinh-1 x = ln (x + ) -∞ < x < ∞ cosh-1 x = ln (x + ) x≥1 [cosh-1 x > 0 is principal value] tanh-1x = ½ln((1 + x)/(1 - x)) -1 < x <1 coth-1 x = ½ln((x + 1)/(x - 1)) x>1 or x < -1 sech-1 x = ln ( 1/x + ) 0 < x ≤ 1 [sech-1 a; > 0 is principal value] csch-1 x = ln(1/x + ) x≠0 Hyperbolic Formulas for Integration u sinh C or ln (u u 2 a 2 ) a a2 u 2 u cosh C or ln (u u 2 a 2 ) a u 2 a2 du du 1 1 du 1 1 au 1 u a 2 u 2 a tanh a C , u a or 2a ln a u C, u a Hyperbolic Formulas for Integration a a2 u2 1 1 1 u ) C, 0 u a u a 2 u 2 a sec h a C or a ln ( u du RELATIONSHIPS OF HYPERBOLIC FUNCTIONS a a2 u2 1 1 1 u ) C , u 0. u a 2 u 2 a csc h a C or a ln ( u du The hyperbolic functions share many properties with the corresponding circular functions. In fact, just as the circle can be represented parametrically by x = a cos t y = a sin t, a rectangular hyperbola (or, more specifically, its right branch) can be analogously represented by x = a cosh t y = a sinh t where cosh t is the hyperbolic cosine and sinh t is the hyperbolic sine. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the equilateral hyperbola. Animated plot of the trigonometric (circular) and hyperbolic functions In red, curve of equation x² + y² = 1 (unit circle), and in blue, x² - y² = 1 (equilateral hyperbola), with the points (cos(θ),sin(θ)) and (1,tan(θ)) in red and (cosh(θ),sinh(θ)) and (1,tanh(θ)) in blue. Animation of hyperbolic functions Applications of Hyperbolic functions Hyperbolic functions occur in the solutions of some important linear differential equations, for example the equation defining a catenary, and 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 arise in many problems of mathematics and mathematical physics in which integrals involving a 2 x 2 arise (whereas the circular functions involve a 2 x 2 ). For instance, the hyperbolic sine arises in the gravitational potential of a cylinder and the calculation of the Roche limit. The hyperbolic cosine function is the shape of a hanging cable (the socalled catenary). The hyperbolic tangent arises in the calculation and rapidity of special relativity. All three appear in the Schwarzschild metric using external isotropic Kruskal coordinates in general relativity. The hyperbolic secant arises in the profile of a laminar jet. The hyperbolic cotangent arises in the Langevin function for magnetic polarization. Derivatives of Hyperbolic Functions d/dx(sinh(x)) = cosh(x) d/dx(cosh(x)) = sinh(x) d/dx(tanh(x)) = sech2(x) Integrals of Hyperbolic Functions ∫ sinh(x)dx = cosh(x) + c ∫ cosh(x)dx = sinh(x) + c. ∫ tanh(x)dx = ln(cosh x) + c. Example : Find d/dx (sinh2(3x)) Sol: Using the chain rule, we have: d/dx (sinh2(3x)) = 2 sinh(3x) d/dx (sinh(3x)) = 6 sinh(3x) cosh(3x) Inverse hyperbolic functions 1 d (sinh−1 (x)) = dx 1 x 2 1 d (cosh−1 (x)) = dx x 2 1 d (tanh−1 (x)) = dx 1 1 x 2 Curves on Roller Coaster Bridge Thank You Animation of a Hypotrochoid Complex Sinh.jpg Sine/Cos Curves