1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

5.3 • COMPLEX EXPONENTS 173

Hence Equation (5-21) becomes

zf =exp [~log(z)] (5-23)

[

1 .8+2mr]

=exp kln(r)+i k

= rtexp(/+:mr) =rt [cos(e+:mr) +isin(

8

+:

1

rn)].

When we again use the periodicity of the complex exponential function, Equation

(5-23) gives k distinct values corresponding to n = 0, 1, ... , k - l. Therefore,

as Example 5.6 illustrated, the fractional power zt is the multivalued kth root

function.

Case (iii): If j and k are positive integers that have no common factors and

c = f, then Equation (5-21) becomes

z•^1 = r•^1 exp [.(8i +2mr)j] k = r• 1[ lcos ((8+2mr)j) k +ism .. ((8+2mr)jk )] ,

and again there are k distinct values that correspond with n = 0, 1, ... , k - l.

Case (iv): If c is not a rational number, then there are infinitely many values

for zc, provided z # 0.

• EXAMPLE 5.8 The values of 2t+~are

zb+IG =exp [ (~ + ;

0

) {ln2 + i2mr)]

=exp [1~2 _ ~; + i (1;~ + 2~n)]

= 2 ~ e-~; [cos (ln2 + 2nn) + isin (In 2 + 2nn)]

50 9 50 9 '

where n is an integer. The principal value of 2i+!b is

2§i +'5!! . = 21 ' [ cos (ln2) 5o +ism .. (ln2)5o ] ""'1.079956 + 0.014972i..

Figure 5.6 shows the terms for this multivalued expression corresponding to

n = - 9, - 8, ... , -1, 0, 1, ... , 8, 9. They exhibit a spiral pattern that is often

present in complex powers.