Chapter 5  Elementary Functions

Overview

    How should complex-valued functions such as  ,  etc., be defined ?  Clearly, any responsible definition should satisfy the following criteria:  

(i)  The functions so defined must give the same values as the corresponding functions for real variables when the number z is a real number.

(ii)  As far as possible, the properties of these new functions must correspond with their real counterparts.  For example, we would want    to be valid regardless of whether z were real or complex.

    These requirements may seem like a tall order to fill.  There is a procedure, however that offers promising results.  It is to put in complex form the expansion of the real functions  ,  etc., as power series into complex form.  We use this strategy in this chapter.

5.1 The Complex Exponential Function

    Recall that the real exponential function can be represented by the power series  .  Thus it is only natural to define the complex exponential  ,  also written as , in the following way:

Definition 5.1 (Exponential Function).  The definition of  exp(z) is  

            .  

Demonstration for Definition 5.1.

    Clearly, this definition agrees with that of the real exponential function when z is a real number.  We now show that this complex exponential has two of the key properties associated with its real counterpart and verify the identity  ,  which, back in Chapter 1 (see identity (1-30) of Section 1.4), we promised to establish.

Theorem 5.1 (The exponential function).  The function    is an entire function satisfying the following conditions:

  (i).   ,  using Leibniz notation  .  

(ii).   ,   i.e.   .  

(iii).   If    is a real number, then   .  

The exponential function is a solution to the differential equation    with the initial condition  .  

Proof of Theorem 5.1.

Demonstration for Theorem 5.1 (i).

Demonstration for Theorem 5.1 (ii).

Demonstration for Theorem 5.1 (iii).

    Note that parts (ii) and (iii) of the Theorem 5.1 combine to verify De Moivre's formula, which we introduced in Section 1.5.

    If  ,  we also see from parts (ii) and (iii) that
    
(5-1)               
            
(5-1)               

Some texts start with this identity as their definition for .  In the exercises, we show that this is a natural approach from the standpoint of differential equations.  

    The notation    is preferred over    in some situations.  For example,    is the value of    when    and equals the positive fifth root of  .  Thus the notation    is ambiguous and might be interpreted as any of the complex fifth roots of the number that we discussed in Section 1.5:

               for  .  

To prevent this confusion, we often use    to denote the single-valued exponential function.

    We now explore some additional properties of  .  Using identity (5-1), we can easily establish that  

(5-2)            ,  for all  z,  provided n is an integer,  

(5-3)            ,  if and only if  ,  where n  is an integer, and

(5-4)            ,  if and only if  ,  for some integer n.  

    For example, because Identity (5-1) involves the periodic functions  cos y  and  sin y,  any two points in the z plane that lie on the same vertical line with their imaginary parts differing by an integral multiple of    are mapped onto the same point in the w plane.  Thus the complex exponential function is periodic with period  ,  which establishes Equation (5-2).  We leave the verification of Equations (5-3) and (5-4) as exercises for the reader.

Example 5.1.  For any integer n, the points    are mapped onto a single point  

              

in the w plane, as indicated in Figure 5.1.

Figure 5.1  The points in the z plane (i.e., the xy plane) and their image   in the w plane (i.e., the uv plane).

Explore Solution 5.1.

    Let's look at the range of the exponential function.  If , we see from identity (5-1) that    can never equal zero, as    is never zero, and the cosine and sine functions are never zero at the same point.  Suppose, then, that  .  If we write w in its exponential form as  ,  identity (5-1) gives
    
            .  

Using identity (5-1), and property (1-39) of Section 1.5 we get

             and , where n is an integer.
Therefore,
            ,
        and
            .  

    Solving these equations for x and y, yields

             and  ,
            
where n is an integer. Thus, for any complex number  ,  there are infinitely many complex numbers such that  .  From the previous equations, we see that the numbers z are  

              
        or
            ,  

where n is an integer. Hence

              

    In summary, the transformation maps the complex plane (infinitely often) onto the set of nonzero complex numbers.
    
    If we restrict the solutions in equation (5-9) so that only the principal value of the argument,  ,  is used, the transformation   maps the horizontal strip  ,  one-to-one and onto the range set  .  This strip is called the fundamental period strip and is shown in Figure 5.2.

            Figure 5.2  The fundamental period strip for the mapping  .  

    The horizontal line  ,  for     in the z plane, is mapped onto the ray     that is inclined at an angle    in the w plane. The vertical segment  ,  for    in the z plane, is mapped onto the circle centered at the origin with radius    in the w plane.  That is,   .

Example 5.2.  Consider a rectangle  , where  .  Show that the transformation    maps the rectangle onto a portion of an annular region bounded by two rays.

Solution.  The image points in the w plane satisfy the following relationships involving the modulus and argument of w:  

            ,   and

            ,  

which is a portion of the annulus    in the w plane subtended by the rays  .   In Figure 5.3, we show the image of the rectangle  

            .  

        Figure 5.3  The image of   under the transformation  .  

Explore Solution 5.2.

Exercises for Section 5.1.  The Complex Exponential Function