ENGR-UH 2610 Fundamentals of Complex Variables
Homework 1
Spring 2024
Due at Recitation Tuesday April 02, 2024
Question 1. For z1 , z2 ∈ C, prove the following:
|z1 − z2 |2 + |z1 + z2 |2 = 2 |z1 |2 + |z2 |2
√
Question 2. If 2z 3 − i 3 = 1, then find z −1 in the standard form x + iy.
Question 3. Sketch the following sets. For each set determine, (a) whether it is a domain? (b)
is it open/closed/neither? (c) is it bounded?
a. 1 < |2z − 6| < 2
b. 0 < Im z < π
c. |z − 1|2 + |z + 1|2 < 8
d. |z − 1| + |z + 1| ≤ 2
e. |Re z| < |z|
Question 4. Express all values of the following expressions in both polar and cartesian coordinates, and plot them.
a. (1 + i)8
b. (−8)1/3
Question 5.
a. Given az 2 + bz + c = 0, (a ̸= 0) where the cofficients a, b and c are complex numbers.
Derive the quadratic formula:
z=
−b + (b2 − 4ac)1/2
2a
where both square roots are to be considered when b2 − 4ac ̸= 0
b. Use the result in part a. to find the roots of the equation
z 2 + 2z + (1 − i) = 0