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676 CHAPTER 19 Complex Numbers and Functions

SECTION 19.1 PROBLEMS

In each of Problems 1 through 10, carry out the indicated 23. Show that, for any positive integer n,
calculation.
4n
i = 1,i 4n+1 = i,i 4n+2 =−1,i 4n+3 =−i.
1. (3 − 4i)(6 + 2i)
2
2
24. Let z = a + ib. Determine Re(z ) and Im(z ).
2. i(6 − 2i) +|1 + i|
25. Show that complex numbers z,w,and u form vertices
3. (2 + i)/(4 − 7i) of an equilateral triangle if and only if
4. ((2 + i) − (3 − 4i))/(5 − i)(3 + i) z + w + u = zw + zu + wu.
2
2
2
5. (17 − 6i)(−4 − 12i)
2
2
26. Show that z = (z) if and only if z is either real or
6. |3i/(−4 + 8i)| pure imaginary.
2
3
7. i − 4i + 2 27. Let z and w be numbers with zw = 1. Suppose either
8. (3 + i) 3 z or w has magnitude 1. Prove that

9. ((−6 + 2i)/(1 − 8i)) 2 z − w = 1.
1 − zw

10. (−3 − 8i)(2i)(4 − i)
Hint: Recall that |u| = uu for every complex num-
2
In each of Problems 11 through 16, determine the magni- ber u.
tude and all of the arguments of z.
28. Show that, for any numbers z and w,
11. 3i 2 2 2 2
|z + w| +|z − w| = 2 |z| +|w| .
12. −2 + 2i
Hint: Note the hint from Problem 27.
13. −3 + 2i
In each of Problems 29 through 37, a set of complex num-
14. 8 + i
bers is specified. Determine whether the set is open, closed,
15. −4 open and closed, or not open and not closed. Specify all
16. 3 − 4i boundary points of the set (whether or not they belong to
the set).
In each of Problems 17 through 22, write the number in
29. M is the set of all z satisfying Im(z)< 7.
polar form.
30. S is the set of all z with |z| > 2.
17. −2 + 2i
31. U is the set of all z with 1 < Re(z) ≤ 3.
18. −7i
32. V is the set of all z with 2 < Re(z) ≤ 3and −1 <
19. 5 − 2i
Imz < 1.
20. −4 − i 2
33. W consists of all z with Re(z)>( Im(z)) .
21. 8 + i
34. R is the set of all numbers 1/m + (1/n)i with m and
22. −12 + 3i n positive integers.
19.2 Complex Functions

A complex function is a function that acts on complex numbers and produces complex
2
numbers. For example, f (z) = z for |z| < 1 is a complex function acting on numbers in
the open unit disk.

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October 15, 2010 18:5 THM/NEIL Page-676 27410_19_ch19_p667-694