The following pages contain detailed solutions of the “Focus-on-Concepts” exercises found in the undergraduate book “A First Course in complex Analysis with Applications” by Dennis G. Zill and Patrick D. Shanahan. We try to make the ideas behind each solution as obvious as possible by elaborating on details such that no missing-links are to be supplied by the reader.
Section 1.1 Focus-on-Concepts 45. What can be said about the complex number z if z=z ? If z2= z 2 ?
46. Think of an alternative solution to Problem 24. Then without doing any significant work, evaluate (1 + i)5404 .
47. For n a nonnegative integer, i n can be one of four values: 1, i, -1, and -i. In each of the following four cases, express the integer exponent n in terms of the symbol k, where k = 0, 1, 2, . . . . (a) i n = 1 (b) i n = i n (c) i = -1 (d) i n = -i
48. There is an alternative to the procedure given in (7). For example, the quotient 56 i /1i must be expressible in the form a + ib: 56 i =aib 1i Therefore, 56i=1iaib . Use this last result to find the given quotient. Use this method to find the reciprocal of 3 − 4i. Solution is mere numerology.
49. Assume for the moment that 1i makes sense in the complex number system. How would you then demonstrate the validity of the equality 1 1 1 1 1i= 2i − 2 2 2 2 2
50. Suppose z1 and z2 are complex numbers. What can be said about z1 or z2 if z1 z2 = 0?
51. Suppose the product z1 z2 of two complex numbers is a nonzero real constant. Show that z 2=k z1 , where k is a real number.
52. Without doing any significant work, explain why it follows immediately from (2) and (3) that z 1 z2 z1 z 2=2 ℜ z1 z2 .
53. Mathematicians like to prove that certain “things” within a mathematical system are unique. For example, a proof of a proposition such as “The unity in the complex number system is unique” usually starts out with the assumption that there exist two different unities, say, 11 and 12 , and then proceeds to show that this assumption leads to some contradiction. Give one contradiction if it is assumed that two different unities exist.
54. Follow the procedure outlined in Problem 53 to prove the proposition “The zero in the complex number system is unique.”
55. A number system is said to be an ordered system provided it contains a subset P with the following two properties: First, for any nonzero number x in the system, either x or −x is (but not both) in P. Second, if x and y are numbers in P, then both xy and x + y are in P. In the real number system the set P is the set of positive numbers. In the real number system we say x is greater than y, written x > y, if and only if x − y is in P . Discuss why the complex number system has no such subset P . [Hint: Consider i and -i.]
