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
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.
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
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 :
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 .
Let us now consider the various possibilities that may arise in the definition of .
Case (i). Suppose
where k an integer. Then,
Recalling that the complex exponential function has period
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.
(ii). If where
k is an integer and , then
Hence Equation (5-21) becomes
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
Case (iii). If
j and k
are positive integers that have no common factors
and , then
Equation (5-21) becomes
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
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.
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,
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,
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
where is any branch of the logarithm whose branch cut does not include the point .
Exercises for Section 5.3. Complex Exponents
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(c) 2006 John H. Mathews, Russell W. Howell