Page 712 - Advanced_Engineering_Mathematics o'neil

692 CHAPTER 19 Complex Numbers and Functions

Rational Powers
A rational number is a quotient of integers. If m and n are positive integers with no common
m 1/n
m
factors, compute z m/n as (z ) , which are the nth roots of z .
EXAMPLE 19.17
3
We will compute all values of (2 − 2i) . Compute (2 − 2i) =−16 − 16i.Wewant thefifth
3/5
√
roots of −16 − 16i. One argument of −16 − 16i is 5π/4 and |− 16 − 16i|= 512, so in polar
form
1/2 i(5π/4+2kπ)
−16 − 16i = (512) e .
The fifth roots of −16 − 16i are
e
(−16 − 16i) 1/5 = (512) 1/10 i(5π/4+2kπ)/5 .
These are the numbers
e
e
(512) 1/10 5πi/4 ,(512) 1/10 13πi/20 ,
e
(512) 1/10 21πi/20 ,(512) 1/10 29πi/20 , and (512) 1/10 37πi/20 .
e
e
Powers z w
If z = 0, define for any complex number w
w
z = e w log(z) .
b
This definition is suggested by the fact that, if a and b are real numbers and a =0, then a =e b ln(a) .
w
If w is not a rational number, then z has infinitely many values.

EXAMPLE 19.18
We will compute all values of (1 − i) 1+i . These numbers are obtained as e (1+i)log(1−i) , so first
√
determine all values of log(1−i).Wehave |1−i|= 2. We also need an argument of 1−i.Any
argument will do. One convenient argument is 7π/4, obtained by a counterclockwise (positive)
rotation from the positive real axis to the point (1,−1). Another argument is −π/4, which is a
clockwise (negative) rotation from the positive real axis to this point. Using the latter, we have
√
1 − i = 2e i(−π/4+2nπ) .
Then all values of log(1 − i) are given by
√

π
ln 2 + i − + 2nπ
4
in which n can be any integer. Every value of (1 − i) 1+i is contained in the expression
√
(1+i)[ ln( 2) +i(−π/4+2nπ)]
e (1+i)log(1−i) = e .
These can be written as
√ √
ln( 2) +π/4−2nπ i( ln( 2) −π/4+2nπ)
e e
√ √

√
= 2e π/4−2nπ cos ln 2 − π/4 + 2nπ + i sin ln 2 − π/4 + 2nπ
√ √

√
= 2e π/4−2nπ cos ln 2 − π/4 + i sin ln 2 − π/4 .

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October 15, 2010 18:5 THM/NEIL Page-692 27410_19_ch19_p667-694