**for**

**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, .

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.

**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.

** 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.

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,

.

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**

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this Mathematica Notebook**

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is**

**Complex Trigonometric
and Hyperbolic Functions**

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