CHAPTER 3 ELEMENTARY FUNCTIONS 28. THE EXPONENTIAL FUNCTION. Definition: The exponential function: The exponential function e z by writing

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1 CHAPTER 3 ELEMENTARY FUNCTIONS We consider here various elementary functions studied in calculus and define corresponding functions of a complex variable. To be specific, we define analytic functions of a complex variable z that reduce to the elementary, functions in calculus when z = x + i0. We start by defining the complex exponential function and then use it to develop the others. 28. THE EXPONENTIAL FUNCTION Definition: The exponential function: The exponential function e z by writing (1) e z = e x e iy (z =x + iy), where Euler's formula (2) e iy = cos y + i sin y is used and y is to be taken in radians. Notes: 1) e z reduces to the usual exponential function in calculus when y = 0. 2) when z = 1/n (n = 2, 3,...). then e z = e l/n is the set of nth roots of e. Properties of e z : 1) z e z z 1 2 = e e 2) z1 e z1 z2 = e z2 e 3) e z is entire and d dx e z z = e. 4) e z # 0 for any complex number z. 5) e z is periodic,with a pure imaginary period 2ni: e z +2nπ = e z. 6) while e x is never negative, there are values of e z that are. 1

2 29. THE LOGARITHMIC FUNCTION Our motivation for the definition of the logarithmic function is based on solving the equation (1) e w = z for w, where z is any nonzero complex number. To do this, we note that when z and w are written z = re iθ (-π < Θ π ) and w = u + i v, the above equation becomes e u e iv = r e iθ then: e u = r and v =Θ+2 n π where n is any integer. That is; u = In r, w has one of the values w = ln r + i(θ +2nπ) (n = 0, ± 1, ± 2,...). Thus, if we write (2) log z = ln r +i(θ + 2nπ) we have the simple relation ( 3 ) e log z = z (z # 0). Note: It is not true that the left-hand side of equation (3) with the order of the exponential and logarithmic functions reversed reduces to just z. More precisely, since expression (2) can be written log z = In z + i arg z and since e z = e x and arg(e z )=y+2nπ ( n = 0, ± l, ± 2,...) when z = x + iy, we know that log(e z ) = In e z + i arg(e z ) = ln(e x ) + i(y + 2nπ) = (x + iy) + 2nπi (n =0, ± 1, ± 2,...). That is, log (e z ) = z + 2nπi, (n =0, ± 1, ± 2,...). 2

3 The principal value of log z is the value obtained from equation (2) when n= 0 there and is denoted by Log z. Thus Log z = ln r + iθ and log z = Log z +2nπi, ( n=0,±1,±2,. ) It reduces to the usual logarithm in calculus when z is a positive real number z = r. 3

4 30. BRANCHES AND DERIVATIVES OF LOGARITHMS If z = re iθ is a nonzero complex number, the argument θ has any one of the values θ = Θ + 2nπ (n = 0, ±1, ±2,...), where Θ = Arg z. Hence the definition (1) log z = ln r + i(θ+2nπi) of the multiple-valued logarithmic function in Sec. 29 can be written (2) log z = ln r + i θ, r > 0, α < θ < α+2π. where α denote any real number. with components u(r,θ)=ln r, v(r,θ)= θ. is single-valued and continuous in the stated domain. Notes: 1) Note that if the function (2) were to be defined on the ray θ = α, it would not be continuous there. 2) The function (2) is not only continuous but also analytic in the domain r > 0, α < θ < α+2π. since the first-order partial derivatives of u and v are continuous there and satisfy the polar form r u r = v θ, u θ = - r v r. of the Cauchy-Riemann equations. 3) Since 4

5 Definition: A branch of a multiple-valued function f is any single-valued function F that is analytic in some domain at each point z of which the value F(z) is one of the values f (z). Observe that, for each fixed α, the single-valued function (2) is a branch of the multiple-valued function (1). The function Log z = ln r + iθ (r>0, -π<θ<π) is called the principal branch. Note that : 1) A branch cut is a portion of a line or curve that is introduced in order to define a branch F of a multiple-valued function. 2) Points on the branch cut for F are singular points of F, and any point that is common to all branch cuts of f is called a branch point. 3) The origin and the ray θ= α make up the branch cut for the branch (2) of the logarithmic function. 4)The branch cut for the principal branch consists of the origin and the ray Θ = π. 5) The origin is evidently a branch point for branches of the multiple-valued logarithmic function. 5

6 31. SOME IDENTITIES INVOLVING LOGARITHMS Let z l and z 2 denote any two nonzero complex numbers, 1) log(z 1 z 2 ) = log z 1 +log z 2. This statement is not, in general, valid when log is replaced everywhere by Log. z 1 2) log = log z1 log z2 z. 2 3) If z is a nonzero complex number, then z n = e n log z, (n=0, ±1,±2, ). 4) It is also true that when z # 0, This property is also true when n is negative integer. 6

7 32. COMPLEX EXPONENTS Definition: When z 0 and the exponent c is any complex number; the function z c is defined by means of the equation z c = e c log z where log z denotes the multiple-valued logarithmic function. EXAMPLE: Properties: 1 1) = z c z 2) d dz z c c = cz c 1 3) The principal value of z c occurs when log z is replaced by Log z. P.V. z c = e c lo g z. 7

8 Definition: The exponential function with base c, where c is any nonzero complex constant, is written c z = e z log c Note that although e z is, in general, multiple-valued the usual interpretation of e z occurs when the principal value of the logarithm is taken. When a value of log c is specified, c z is an entire function of z. In fact, d z z c = c logc. dz Examples : 8

9 33. TRIGONOMETRIC FUNCTIONS Definition : 1) The sine and cosine functions of a complex variable z is defined as follows: iz iz iz iz e e e + e sin z =, cos z =. 2i 2 2) These functions are entire since they are linear combinations of the entire functions e iz and e -iz. 3) Knowing the derivatives of those exponential functions, we find that d d sin z = cos z, cos z = sin z dz dz 4) It is easy to see from definitions (1) that sin(-z) = - sin z and cos(-z) = cos z; A variety of other identities from trigonometry are valid with complex variables. 5) sin(z 1 + z 2 ) = sin z 1 cos z 2 + cos z 1 sin z 2, 6) cos(z 1 + z 2 ) = cos z l cos z 2 - sin z l sin z 2 ; 7) sin 2 z + cos 2 z = 1, 8) sin 2z = 2 sin z cos z, cos2z = cos 2 z-sin 2 z, 9) sin(z+π/2) =cos z, sin (z π/2) = - cos z. 10) sin z = sin x cosh y + i cos x sinh y 11) cos z = cos x cosh y - i sin x sinh y, 12) sin(z+2π)=sin z, sin(z+π)=-sin z; 9

10 13) cos(z+2π)=cos z, cos(z+π)=-cos z; 14) sin z 2 = sin 2 x + sinh 2 y; 15) cos z 2 = cos 2 x + sinh 2 y; 16) from the last two equations that sin z and cos z are not bounded on the complex plane, where as the absolute values of sin x and cos x are less than or equal to unity for all values of x. 17) A zero of a given function f (z) is a number z 0 such that f(z 0 )=0. Thus: and Definition of the other trigonometric functions: Note that tan z and sec z are analytic everywhere except at the singularities While cot z and csc z have singularities at the zeros of sin z namely; Their derivatives are: 10

11 The period of each of these trigonometric functions follows from their definitions, that is; tan(z+π) = tan z, cot(z+π) = cot z, while; sec(z+2π) = sec z, csc(z+2π) = csc z. 11

12 34. HYPERBOLIC FUNCTIONS The hyperbolic sine and the hyperbolic cosine of a complex variable are defined as they are with a real variable; that is, Since e Z and e -Z are entire, it follows from definitions (1) that sinh z and cosh z are entire. Furthermore The hyperbolic sine and cosine functions are closely related to those trigonometric functions: Some of the most frequently used identities involving hyperbolic sine and cosine functions are Example: 12

13 The period of sinh z and cosh z is 2πi and Differentiation Formulas: 13

14 35. INVERSE TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS Inverses of the trigonometric and hyperbolic functions can be described in terms of logarithms. In order to define the inverse sine function sin -1 z, we write w = sin -1 z when z = sin w. That is, w = sin -1 z when Example: 14

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