1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

1 74 CHAPTER 5 • ELEMENTARY FUNCTIONS

)'

3

•^2 •

I •

• • •

•

-2 -I • •

• •

2 3

• •

-I

• •

-2

Figure 5. 6 Some of the values of 2 t+ ~.

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 z f:. 0, then

where n is an integer.

(5- 24)

(5-25)

(5- 26)

(5- 27)

The following example shows that Identity (5-27) does not hold if n is re-

placed with an arbitrary complex value.

• EXAMPLE 5.9
  (i^2 ); = exp[ilog(- 1)] = c (1+^2 nl", wher e n is an integer, and
  (i)^2 ' =exp (2ilogi) = e-(1+^4 n)" , where n is an integer.

Since these sets of solutions are not equal, Identity (5- 27) does not always hold.

We can compute the derivative of the principal branch of z<, which is t he

function f'(z) =exp [cLog (z)]. By the chain rule,

f' (z) = ~exp [cLog (z)]. (5-28)

z