The Complex Exponential Function
Chapter 5 Elementary Functions
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
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:
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
(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)
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
Using identity (5-1), and property
(1-39) of Section
1.5 we get
and , where n is an integer.
Solving these equations for x and y, yields
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
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, .
5.2. Consider a rectangle ,
where . Show
that the transformation maps
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:
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
History of Complex Numbers
The Next Module is
The Complex Logarithm Function
Return to the Complex Analysis Modules
Return to the Complex Analysis Project
This material is coordinated with our book Complex Analysis for Mathematics and Engineering.
(c) 2012 John H. Mathews, Russell W. Howell