1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

5. 3 • COMPLEX EXPONENTS 171

Definition 5.4: Complex exponent

Let c be a complex number. We define zc as

zc =exp [clog (z)]. (5-21)

The right side of Equation (5- 21 ) is a set. This definition makes sense because
if both z and care real numbers with z > 0, Equation (5-21) gives the familiar
(real) definition for zc, as t he following example illustra tes.

• EXAMPLE 5.6 Use Equation (5-21) to evaluate 4!.

Solution Calculating 4t =exp[~ log(4)] gives

~ log(4) = {ln2 + in1f: n is an integer}.

Thus, 4! is the set {exp (In 2 + in'lr) : n is an integer}. The distinct values occur
when n = 0 and 1; we get exp (ln 2) = 2 and exp (ln2 + i7r) =exp (In 2) exp (i7r) =
-2. In other words, d = {- 2, 2}.

The expression 4 ~ is different from J4, as the former represents the set

{- 2, 2} and the latter gives only one value, J4 = 2.

Because log(z) is multivalued, the function zc will, in general, be multival-
ued. If we want to focus on a single value for zc, we can do so via the function
defined for z ':f 0 by

f (z) =exp (cLog (z)], (5-22)

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

• EXAMPLE 5. 7 Find the principal values of v l + i and i;.

Solution From Example 5.3,

. ln2 ,1f! .1f
Log(l+i) =
2

+ i

4

=In2•+i

4

and

Log (i) = i~.