Integration of Hyperbolic Functions

6 Key excerpts on "Integration of Hyperbolic Functions"

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  Calculus of a Single Variable: Early Transcendental Functions, International Metric Edition

  • Ron Larson, Bruce Edwards(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Differentiate and integrate functions involving inverse hyperbolic functions. Hyperbolic Functions In this section, you will look briefly at a special class of exponential functions called hyperbolic functions. The name hyperbolic function arose from comparison of the area of a semicircular region, as shown in Figure 5.46, with the area of a region under a hyperbola, as shown in Figure 5.47. x - 1 1 2 y = 1 - x 2 y x - 1 1 2 y y = 1 + x 2 Circle: x 2 + y 2 = 1 Hyperbola: -x 2 + y 2 = 1 Figure 5.46 Figure 5.47 The integral for the semicircular region involves an inverse trigonometric (circular) function: integral.alt1 1 -1 radical.alt21 - x 2 dx = 1 2 bracketleft.alt2 xradical.alt21 - x 2 + arcsin x bracketright.alt2 1 -1 = π 2 ≈ 1.571. The integral for the hyperbolic region involves an inverse hyperbolic function: integral.alt1 1 -1 radical.alt21 + x 2 dx = 1 2 bracketleft.alt2 xradical.alt21 + x 2 + sinh -1 x bracketright.alt2 1 -1 ≈ 2.296. This is only one of many ways in which the hyperbolic functions are similar to the trigonometric functions. Definitions of the Hyperbolic Functions sinh x = e x - e -x 2 csch x = 1 sinh x , x ≠ 0 cosh x = e x + e -x 2 sech x = 1 cosh x tanh x = sinh x cosh x coth x = 1 tanh x , x ≠ 0 FOR FURTHER INFORMATION For more information on the development of hyperbolic functions, see the article “An Introduction to Hyperbolic Functions in Elementary Calculus” by Jerome Rosenthal in Mathematics Teacher. To view this article, go to MathArticles.com. REMARK The notation sinh x is read as “the hyperbolic sine of x,” cosh x as “the hyperbolic cosine of x,” and so on. JOHANN HEINRICH LAMBERT (1728–1777) The first person to publish a comprehensive study on hyperbolic functions was Johann Heinrich Lambert, a Swiss- German mathematician and colleague of Euler. See LarsonCalculus.com to read more of this biography. AIP Emilio Segre Visual Archives, Physics Today

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  Engineering Mathematics

  A Programmed Approach, 3th Edition

  • C W. Evans, C. Evans(Authors)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  Hyperbolic functions 5

  Although we have now explored some of the basic terminology of mathematics and developed the techniques of the differential calculus, we need to pause to extend our algebraic knowledge. In this chapter we shall describe a class of functions known as the hyperbolic functions which are very similar in some ways to the circular functions. We shall use the opportunity to consider in detail what is meant by an inverse function.

  After studying this chapter you should be able to □  Use the hyperbolic functions and their identities; □  Solve algebraic equations which involve hyperbolic functions; □  Differentiate hyperbolic functions; □  Decide when a function has an inverse function; □  Express inverse hyperbolic functions in logarithmic form. We shall also consider a practical problem concerning the sag of a chain. 5.1      DEFINITIONS AND IDENTITIES

  The hyperbolic functions are in some ways very similar to the circular functions. Indeed when we deal with complex numbers (Chapter 10 ) we shall see that there is an algebraic relationship between the two. Initially we shall discuss the hyperbolic functions algebraically, but later we shall see that one of them arises in a physical context.

  The functions cosine and sine are called circular functions because x = cos θ and y = sin θ satisfy the equation x2 + y2 = 1, which is the equation of a circle. The functions known as the hyperbolic cosine (cosh) and the hyperbolic sine (sinh) are called hyperbolic functions because x = cosh u and y = sinh u satisfy the equation x2 – y2 = 1, which is the equation of a rectangular hyperbola.

  We shall define the hyperbolic functions and use these definitions to sketch their graphs. Here then are the definitions:

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  What is Calculus?

  From Simple Algebra to Deep Analysis

  • R Michael Range(Author)
  • 2015(Publication Date)
  • WSPC

    (Publisher)

  Taylor’s Theorem with Remainder . As an application, we then verify the convergence statements we had made in Section III.9, and we shall place these results in a broader context.

  IV.9.1     An Application of Integration by Parts

  Suppose f is an algebraic function, or more generally, a function whose derivatives of any order exist on an interval I . The derivatives of f are then continuous and hence bounded on any closed bounded interval contained in I . Therefore, by Theorem 6.2 , any such derivative is then integrable. Fix a point a ∈ I . Since f is an antiderivative of D (f ), the Fundamental Theorem of Calculus implies that

  We now fix x and apply integration by parts to the integral on the right side, in the form ∫ f ′dg = f ′ g − ∫ g d (f ′), where we choose g so that dg = 1dt , that is, so that g ′ = 1. Hence g (t ) = t + c , where we are free to choose the constant c . It follows that

  The right side is an antiderivative of f ′ on the left. By the Fundamental Theorem of Calculus (Part 2) one therefore obtains

  In order to simplify this expression, we choose c = −x , so that the first term vanishes. (Remember: we treat x as constant; differentiation and integration are with respect to t !) We eliminate the minus signs in the preceding formula by interchanging the order in the resulting terms (a − x ) and (t − x

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  Functions Mathematics

  • (Author)
  • 2014(Publication Date)
  • Learning Press

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter 6 Hyperbolic Function & Trigonometric Functions Hyperbolic function A ray through the origin intercepts the hyperbola in the point , where is twice the area between the ray and the -axis. For points on the hyperbola below the -axis, the area is considered negative . ________________________ WORLD TECHNOLOGIES ________________________ In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The inverse hyperbolic functions are the area hyperbolic sine arsinh (also called asinh, or sometimes by the misnomer of arcsinh) and so on. 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. 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 take real values for a real argument called a hyperbolic angle. In complex analysis, they are simply rational functions of exponentials, and so are meromorphic. Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert. Riccati used Sc. and Cc. ( [co]sinus circulare ) to refer to circular functions and Sh. and Ch. ( [co]sinus hyperbolico ) to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what they are today. Standard algebraic expressions sinh, cosh and tanh ________________________ WORLD TECHNOLOGIES ________________________ csch, sech and coth The hyperbolic functions are: • Hyperbolic sine: • Hyperbolic cosine: • Hyperbolic tangent: • Hyperbolic cotangent:

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  Calculus

  Late Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  400 Chapter 6 / Exponential, Logarithmic, and Inverse Trigonometric Functions HYPERBOLIC IDENTITIES The hyperbolic functions satisfy various identities that are similar to identities for trigono- metric functions. The most fundamental of these is cosh 2 x − sinh 2 x = 1 (1) which can be proved by writing cosh 2 x − sinh 2 x = (cosh x + sinh x)(cosh x − sinh x) =  e x + e −x 2 + e x − e −x 2   e x + e −x 2 − e x − e −x 2  = e x · e −x = 1 Other hyperbolic identities can be derived in a similar manner or, alternatively, by per- forming algebraic operations on known identities. For example, if we divide (1) by cosh 2 x, we obtain 1 − tanh 2 x = sech 2 x and if we divide (1) by sinh 2 x, we obtain coth 2 x − 1 = csch 2 x The following theorem summarizes some of the more useful hyperbolic identities. The proofs of those not already obtained are left as exercises. 6.8.2 THEOREM cosh x + sinh x = e x sinh (x + y) = sinh x cosh y + cosh x sinh y cosh x − sinh x = e −x cosh (x + y) = cosh x cosh y + sinh x sinh y cosh 2 x − sinh 2 x = 1 sinh (x − y) = sinh x cosh y − cosh x sinh y 1 − tanh 2 x = sech 2 x cosh (x − y) = cosh x cosh y − sinh x sinh y coth 2 x − 1 = csch 2 x sinh 2x = 2 sinh x cosh x cosh (−x) = cosh x cosh 2x = cosh 2 x + sinh 2 x sinh (−x) = −sinh x cosh 2x = 2 sinh 2 x + 1 = 2 cosh 2 x − 1 WHY THEY ARE CALLED HYPERBOLIC FUNCTIONS Recall that the parametric equations x = cos t, y = sin t (0 ≤ t ≤ 2π ) represent the unit circle x 2 + y 2 = 1 (Figure 6.8.3a), as may be seen by writing Figure 6.8.3 x 2 + y 2 = cos 2 t + sin 2 t = 1 If 0 ≤ t ≤ 2π , then the parameter t can be interpreted as the angle in radians from the positive x-axis to the point (cos t, sin t) or, alternatively, as twice the shaded area of the sector in Figure 6.8.3a (verify).

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  Calculus

  Single Variable

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  This similarity is somewhat surprising, since there is little on the surface to suggest that there should be any relationship between exponential and trigonometric functions. This is because the relationship occurs within the context of complex numbers, a topic we will leave for more advanced courses. Definitions of Hyperbolic Functions To introduce the hyperbolic functions, observe that the function e x can be expressed in the follow- ing way as the sum of an even function and an odd function: e x = e x + e −x 2    Even + e x − e −x 2    Odd These functions are sufficiently important that there are names and notation associated with them: the odd function is called the hyperbolic sine of x and the even function is called the hyperbolic cosine of x. They are denoted by sinh x = e x − e −x 2 and cosh x = e x + e −x 2 where sinh is pronounced “cinch” and cosh rhymes with “gosh.” From these two building blocks we can create four more functions to produce the following set of six hyperbolic functions. The terms “tanh,” “sech,” and “csch” are pronounced “tanch,” “seech,” and “coseech,” respectively. Definition 6.8.1 Hyperbolic sine sinh x = e x − e −x 2 Hyperbolic cosine cosh x = e x + e −x 2 Hyperbolic tangent tanh x = sinh x cosh x = e x − e −x e x + e −x Hyperbolic cotangent coth x = cosh x sinh x = e x + e −x e x − e −x Hyperbolic secant sech x = 1 cosh x = 2 e x + e −x Hyperbolic cosecant csch x = 1 sinh x = 2 e x − e −x 6.8 Hyperbolic Functions and Hanging Cables 393 Example 1 sinh 0 = e 0 − e −0 2 = 1 − 1 2 = 0 cosh 0 = e 0 + e −0 2 = 1 + 1 2 = 1 sinh 2 = e 2 − e −2 2 ≈ 3.6269 Technology Mastery Compu- ter algebra systems have built-in capabilities for eval- uating hyperbolic functions directly, but some calcula- tors do not. However, if you need to evaluate a hyperbolic function on a calculator, you can do so by expressing it in terms of exponential func- tions, as in Example 1. Jeremy Edwards/iStockphoto The design of the Gateway Arch in St.

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