Complex Hyperbolic Functions

3 Key excerpts on "Complex Hyperbolic Functions"

• 

  eBook - ePub

  Engineering Mathematics

  A Programmed Approach, 3th Edition

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

    (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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  eBook - ePub

  Analysis And Mathematical Physics

  • Shaun Bullett, Tom Fearn;Frank Smith;;(Authors)
  • 2016(Publication Date)
  • WSPC (EUROPE)

    (Publisher)

  This chapter is based on material covered in part of my London Taught Course Centre course on Complex Analysis. The other half concerned the value distribution of meromorphic functions. Special functions are natural generalisations of elementary functions. They are usually considered “special” because they satisfy a number of identities, including differential or difference equations. They also have simple singularity structure when considered in the complex domain.

  Elliptic functions are natural generalisations of trigonometric functions. They appear in many applications such as the equations of motion for a simple pendulum and in the calculation of the perimeters of several simple geometric figures including ellipses (hence the name “elliptic”) and lemniscates. We will describe elliptic functions in Section 2 . We will concentrate on the complex analytic theory of these functions as we will not have space to describe, e.g. their importance in number theory.

  In Section 3 we will study Fuchsian linear differential equations. We will mostly be concerned with the hypergeometric equation. Various limits of this equation in which singularities merge include many of the important classical equations of mathematical physics: the confluent hypergeometric, Bessel, Hermite and Airy equations. The general solution of the hypergeometric equation is branched at 0, 1 and ∞. We will describe the analytic continuation of solutions and construct the monodromy matrices that describe how a basis of solutions changes as it is continued along closed paths. In Section 4 we will study matrix Fuchsian equations. We will show that demanding that the monodromy data remains fixed as we vary the locations of the singularities results in the Schlesinger equations. These are important integrable nonlinear equations. In the case of four regular singularities, the resulting Schlesinger equation is equivalent to the sixth Painlevé equation.

  2. Elliptic Functions

  Two fundamental properties of trigonometric functions are that they are meromorphic and periodic. An elliptic function is a meromorphic function that has two independent periods.

  Definition 2.1. Let ω1 and ω2 be two complex numbers whose ratio is not real. Then any function f for which

  for all z in the domain of f is said to be doubly periodic with periods 2ω1 and 2ω2 . If f is doubly periodic and meromorphic (in all C ) then it is said to be an elliptic function

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  eBook - ePub

  A Journey into the World of Exponential Functions

  • Gautam Bandyopadhyay(Author)
  • 2023(Publication Date)
  • CRC Press

    (Publisher)

  β − i α ) = cos β cos i α + sin β sin i α = cos β cos h α + i sin β sinh α

  Similarly,

  (4.39)

  ∴ sin

  (

  β − i α

  )

  = sin β cos i α + cos β sin i α = sin β cos h α − i sin β sinh α

  equation (4.38) and equation (4.39) show that trigonometric functions of a complex angle are complex quantities of the form a ± ib.

  Now we will write similar expressions for hyperbolic sine and cosine functions.

  cosh

  (

  α + i β

  )

  = cosh α cos i β + sinh α sinh i β

  Now,

  (4.40)

  cosh i β =

  e

  i β

  +

  e

  − i β

  2

  = cos β     a n d sinh i β =

  e

  i β

  +

  e

  − i β

  2

  = i sin β

  ∴ cosh

  (

  α + i β

  )

  = cosh α cos β + i sinh α sin β

  Again,

  (4.41)

  sinh

  (

  α + i β

  )

  = sinh α cosh i β + cosh α sinh i β

  = sinh α cos β + i cosh α sin β

  Equation (4.40) and equation (4.41) show that hyperbolic functions of a complex angle are also complex quantities of the form a ± ib.

  4.13 Applications

  Many natural and physical processes can be described by differential equations; very often we find that functions like A1 emt + A2 e– mt satisfy those equations where A1 and A2 are constants. Here m may be a real number or an imaginary number or a complex number. Two exponential functions with real exponents may produce a hyperbolic function indicating increase or decrease in the magnitude of a certain quantity. Two exponential functions with imaginary exponents mayproduce an oscillation which can be mathematically described by cosine (or sine) type of circular functions. Two exponential functions with complex conjugate numbers as their exponents may produce a rising or decaying oscillation.

  4.13.1 A DC Voltage Source (V) is switched on to an LC circuit

  We now refer to Figure 4.12 and write down the following equation.

  Figure 4.12

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