Hyperbolic functions dfs

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This ppt. is about what Hyperbolic functions and curves are and where we use them in daily life.

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Hyperbolic functions dfs

  1. 1. Hyperbolic FunctionsDr. Farhana ShaheenYanbu University CollegeKSA
  2. 2. Hyperbolic Functions Vincenzo Riccati (1707 - 1775) is given credit for introducing the hyperbolic functions. Hyperbolic functions are very useful in both mathematics and physics.
  3. 3. The hyperbolic functions are: Hyperbolic sine: Hyperbolic cosine
  4. 4. Equilateral hyperbola x = coshα , y = sinhα x2 – y2= cosh2 α - sinh2 α = 1.
  5. 5. GRAPHS OF HYPERBOLICFUNCTIONS y = sinh x y = cosh x
  6. 6. Graphs of cosh and sinh functions
  7. 7. The St. Louis arch is in the shape of ahyperbolic cosine.
  8. 8. Hyperbolic Curves
  9. 9. y = cosh x The curve formed by a hanging necklace is called a catenary. Its shape follows the curve of y = cosh x.
  10. 10. Catenary Curve 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 a electric wire hanging between two telephone poles. 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.
  11. 11. Catenary Curve The curve is described by a COSH(theta) function
  12. 12. Example of non-catenary curves
  13. 13. Sinh graphs
  14. 14. Graphs of tanh and coth functions y = tanh x y = coth x
  15. 15. Graphs of sinh, cosh, and tanh
  16. 16. Graphs of sech and csch functions y = sech x y = csch x
  17. 17.  Useful relations   Hence: 1 - (tanh x)2 = (sech x)2.    
  18. 18. RELATIONSHIPS OF HYPERBOLICFUNCTIONS 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
  19. 19.  The following list shows the principal values of the inverse hyperbolic functions expressed in terms of logarithmic functions which are taken as real valued.
  20. 20.  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
  21. 21. Hyperbolic Formulas for Integration du 1 u 2 2 sinh C or ln ( u u a ) 2 2 a u a du 1 u 2 2 cosh C or ln ( u u a ) 2 2 u a a du 1 1 u 1 a u 2 2 tanh C,u a or ln C, u a a u a a 2a a u
  22. 22. Hyperbolic Formulas for Integration 2 2 du 1 1 u 1 a a u sec h C or ln ( ) C,0 u a 2 2 u a u a a a uRELATIONSHIPS OF HYPERBOLIC FUNCTIONS 2 2 du 1 1 u 1 a a u csc h C or ln ( ) C, u 0. 2 2 u a u a a a u
  23. 23.  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.
  24. 24.  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.
  25. 25. 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.
  26. 26. Animation of hyperbolic functions
  27. 27. Applications of Hyperbolic functions Hyperbolic functions occur in the solutions of some important linear differential equations, for example the equation defining a catenary, and Laplaces equation in Cartesian coordinates. The latter is important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.
  28. 28.  The hyperbolic functions arise in many problems of mathematics and mathematical physics in which integrals involving a x arise (whereas the 2 2 circular functions involve a x 2 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 so- called catenary).
  29. 29.  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.
  30. 30. Derivatives of Hyperbolic Functions d/dx(sinh(x)) = cosh(x) d/dx(cosh(x)) = sinh(x) d/dx(tanh(x)) = sech2(x)
  31. 31. Integrals of Hyperbolic Functions ∫ sinh(x)dx = cosh(x) + c ∫ cosh(x)dx = sinh(x) + c. ∫ tanh(x)dx = ln(cosh x) + c.
  32. 32. 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)
  33. 33. Inverse hyperbolic functions d (sinh−1 (x)) = 1 2 dx 1 x d 1 (cosh−1 (x)) = 2 dx x 1 d 1 (tanh−1 (x)) = 2 dx 1 x
  34. 34. Curves on Roller Coaster Bridge
  35. 35. Masjid in Kazkhistan
  36. 36. Fatima masjid in Kuwait
  37. 37. Kul Sharif Masjid in Russia
  38. 38. Masjid in Georgia
  39. 39. Great Masjid in China
  40. 40. Thank You
  41. 41. Animation of a Hypotrochoid
  42. 42. Complex Sinh.jpg

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