Module

for

Complex Exponents and Complex Powers

5.3  Complex Exponents

In Section 1.5 we indicated that it is possible to make sense out of expressions such as or without appealing to a number system beyond the framework of complex numbers.  We now show how this is done by taking note of some rudimentary properties of the complex exponential and logarithm, and then using our imagination.

We begin by generalizing Identity (5-15).  Equations (5-12) and (5-14) show that can be expressed as the set .  We can easily show (left as an exercise) that, for , , where is any branch of the function .  But this means that for any , the identity holds true.  Because denotes the set , we see that , for .

Next, we note that identity (5-17) gives us , where n is any natural number, so that for .  With these preliminaries out of the way, we can now come up with a definition of a complex number raised to a complex power.

Definition 5.4.  Let c be a complex number.  We define as follows

(5-21)            .

The right side of Equation (5-21) is a set.  This definition makes sense because, if both  z  and  c  are real numbers with  , Equation (5-21) gives the familiar (real) definition for  , as the following example illustrates.

Example 5.6.  Use Equation (5-21) to evaluate .

Solution.  Calculating    gives

.

Thus    is the set  .  The distinct values occur when ;  we get    and  .  In other words, .

Remark.  The expression    is different from  ,  as the former represents the set    and the latter gives only one value,  .

Explore Solution 5.6.

Because is multivalued, the function will, in general, be multivalued.  If we want to focus on a single value for , we can do so via the function defined for by

(5-22)

which is called the principal branch of the multivalued function  .  Note that the principal branch of    is obtained from Equation (5-21) by replacing    with the principal branch of the logarithm.

Example 5.7. Find the principal value of  (a)   ,  and  (b)  .

Solution. From Example 5.3,

Identity (5-22) yields the principal values of and :

and

Note that the result of raising a complex number to a complex power may be a real number in a nontrivial way.

Explore Solution 5.7 (a).

Explore Solution 5.7 (b).

Figure 5.C  The mapping  .

Complex Exponents

Let us now consider the various possibilities that may arise in the definition of  .

Case (i).  Suppose where k an integer.  Then, if  ,

.

Recalling that the complex exponential function has period , we have

.

which is the single-valued kth power of z that we discussed in Section 1.5.  This is easily verified by the computation

which is the single-valued kth power of z that we discussed in Section 1.5.

Case (ii).  If    where k is an integer and  ,  then

(5-23)            .

Hence Equation (5-21) becomes

for  .

When we again use the periodicity of the complex exponential function, Equation (5-23) gives k distinct values corresponding to  .  Therefore, as Example 5.6 illustrated, the fractional power is the multivalued root function.  Equation (5-23) is easily verified by the computation

Case (iii).  If j and k are positive integers that have no common factors and  ,  then Equation (5-21) becomes

for  .

This is easy to establish.  If    then

and again there are k distinct values corresponding to  .

Case (iv).  Suppose c is not a rational number, then there are infinitely many values for  ,  provided  .

Example 5.8.  The values of of    are

where n is an integer.  The principal value of    is

Figure 5.6 shows the terms for this multivalued expression corresponding to  .  They exhibit a spiral pattern that is often present in complex powers.

Explore Solution 5.8.

Figure 5.6  Some of the "spiral pattern" of values for  .

The Rules for exponents.

Some of the rules for exponents carry over from the real case.  In the exercises we ask you to show that if c and d are complex numbers and  ,  then

where n is an integer.

The following example shows that Identity (5-21) does not hold if n is replaced with an arbitrary complex value.

Example 5.9.  (a)  ,

and  (b) .

Since these sets of solutions are not equal, Identity (5-21)  does not always hold.

Explore Solution 5.9.

We can compute the derivative of the principal branch of , which is the function

(5-28)            .   By the chain rule,
(5-28)
(5-28)            .

If we restrict to the principal branch,  , then Equation (5-28) can be written in the familiar form that you learned in calculus.  That is, for and z not a negative real number,

.

Exploration

We can use Identity (5-21) to define the exponential function with base b, where is a complex number:

.

If we specify a branch of the logarithm, then will be single-valued and we can use the rules of differentiation to show that the resulting branch of is an analytic function.  The derivative of is then given by the familiar rule

(5-29)

where is any branch of the logarithm whose branch cut does not include the point .

Exercises for Section 5.3.  Complex Exponents

Exponential Function

Complex Logarithms

The Next Module is

Complex Trigonometric and Hyperbolic Functions

(c) 2006 John H. Mathews, Russell W. Howell