The Real And Complex Number Systems

1.39 State and prove a theorem analogous to Theorem 1.54, expressing
Log (z 1 /z 2 ) in terms of Log (z 1 ) and Log (z 2 ) .
Proof: Write z 1 = r 1 e i arg(z 1) and z 2 = r 2 e i arg(z 2) , then
Hence,
∣ Log (z 1 /z 2 ) = log
z 1 ∣∣∣
∣ + i arg
z 2
z 1
z 2
= r 1
r 2
e i[arg(z 1)−arg(z 2 )] .
(
z1
z 2
)
= log |z 1 | − log |z 2 | + i [arg (z 1 ) − arg (z 2 ) + 2πn (z 1 , z 2 )] by xercise 1.38
= Log (z 1 ) − Log (z 2 ) + i2πn (z 1 , z 2 ) .
1.40 Prove that the nth roots of 1 (also called the nth roots of unity)
are given by α, α 2 , ..., α n , where α = e 2πi/n , and show that the roots ≠ 1
satisfy the equation
1 + x + x 2 + ... + x n−1 = 0.
Proof: By Theorem 1.51, we know that the roots of 1 are given by
α, α 2 , ..., α n , where α = e 2πi/n . In addition, since
which implies that
x n = 1 ⇒ (x − 1) ( 1 + x + x 2 + ... + x n−1) = 0
1 + x + x 2 + ... + x n−1 = 0 if x ≠ 1.
So, all roots except 1 satisfy the equation
1 + x + x 2 + ... + x n−1 = 0.
1.41 (a) Prove that |z i | < e π for all complex z ≠ 0.
Proof: Since
z i = e iLog(z) = e − arg(z)+i log|z| ,
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