The Real And Complex Number Systems

1.43 (a) Prove that Log (z w ) = wLog z +2πin.
Proof: Write w = u + iv, where u and v are real. Then
Log (z w ) = log |z w | + i arg (z w )
= log [ e u log|z|−v arg(z)] + i [v log |z| + u arg (z)] + 2πin by Exercise1.42
= u log |z| − v arg (z) + i [v log |z| + u arg (z)] + 2πin.
On the other hand,
wLogz + 2πin = (u + iv) (log |z| + i arg (z)) + 2πin
= u log |z| − v arg (z) + i [v log |z| + u arg (z)] + 2πin.
Hence, Log (z w ) = wLog z +2πin.
Remark: There is another proof by considering
e Log(zw) = z w = e wLog(z)
which implies that
Log (z w ) = wLogz + 2πin
for some n ∈ Z.
(b) Prove that (z w ) α = z wα e 2πinα , where n is an integer.
Proof: By (a), we have
(z w ) α = e αLog(zw) = e α(wLogz+2πin) = e αwLogz e 2πinα = z αw e 2πinα ,
where n is an integer.
1.44 (i) If θ and a are real numbers, −π < θ ≤ π, prove that
(cos θ + i sin θ) a = cos (aθ) + i sin (aθ) .
Proof: Write cos θ + i sin θ = z, we then have
(cos θ + i sin θ) a = z a = e aLogz = e a[log|eiθ |+i arg(e iθ )] = e
iaθ
= cos (aθ) + i sin (aθ) .
29