WA 1
April 3, 2009
Submit your solutions on Thursday, April 9.
(2 pts.) Problem 1. Find the real and imaginary parts of:
2
1−3i,(1 + i√3)6,1 + i
1−i5
, 1 + i√3
1−i!4
.
(2 pts.) Problem 2. Find all complex zfor which (1 +z)(1 −z)−1is: (i) purely
real; (ii) purely imaginary.
(2 pts.) Problem 3. Prove the identity
|z1+z2|2+|z1−z2|2= 2(|z1|2+|z2|2)
and explain its geometrical meaning.
(1 pt.) Problem 4. Show that if ω1, . . . , ωnare the n-th roots of z06= 0 then
ω1+· ·· +ωn= 0.
(2 pts.) Problem 5. Show that for any complex w6= 0 and any real αthe
equation ez=whas exactly one solution zsatisfying α < Im z≤α+ 2π.
(1 pt.) Problem 6. For which zis the exponential function (i) purely real; (ii)
purely imaginary?
1
