C S U , B

C ALIFORNIA S TATE U NIVERSITY, B AKERSFIELD (CSUB)
D EPARTMENT OF E LECTRICAL & C OMPUTER E NGINEERING & C OMPUTER S CIENCE
ECE 306: C OMPLEX A NALYSIS IN E NGINEERING
Homework 2
Solution
QUESTION 1:(20 POINTS)
a) Write the following numbers in the form of a + bi .
1. e −i π/4
2.
e 1+i 3π
e −1+i π/2
b) Write the following numbers in the polar form.
1.
1−i
3
p
2. −8π(1 + 3i )
3. (1 + i )6
c) Show that for all z:
1. e z+πi = −e z
2. e z = e z
d) Determine which of the following properties of the real exponential function remain true
for the complex exponential function, ( i.e., for x replaced by z).
1. e x is never zero.
2. e x is a one-to-one function.
3. e x is defined for all x.
4. e −x =
1
ex .
Answer
p
p
a) 1. 22 − 22 i
1
2. e 2 i
b) p
1. 32 e −i π/4
2. 16πe −i 2π/3
3. 8e i 3π/2
c)
1. e z+πi = e x [cos(y + π) + i sin(y + π)] = −e x [cos(y) + i sin(y)] = −e z
2. e z = e x cis(y) = e x (cos(y) − i sin(y)) = e x cos(−y) + i sin(−y)) = e z
d) a,c, and d are true. b is false because e z+2πi = e z . So, the complex exponential function is
not one-to-one and is in fact periodic with period 2πi .
2
QUESTION 2:(10 POINTS)
a) Find all the values of the following numbers:
1. (−16)1/4
2. i 1/4
3. (i − 1)1/2
b) What are the roots of the equations below.
1. z 2 − 2z + i = 0
2. z 4 + 1 = 0
Answer
a)
π+2kπ
1. 2e i 4 , k = 0, 1, 2, 3
π/2+2kπ
2. e i 4 , k = 0, 1, 2, 3
3π/4+2kπ
3. 21/4 e i 2 , k = 0, 1
b)
p
¡
¢
1. z = 1 ± 1 − i = 1 ± 21/4 cos(pi /8) − i sin(pi /8)
2. z = (−1)1/4 = e (i
π+2kπ
)
4
, k = 0, 1, 2, 3
3
QUESTION 3:(20 POINTS) Refer to the sets described by the following inequalities:
a) 0 < |z − 2| < 3
b) −1 < Im(z) ≤ 1
c) |z| ≥ 2
d)
¡
¢2
Re(z) > 1
1. Which of the given sets are open?
2. Which of the given sets are domains?
3. Which of the given sets are bounded?
4. Which of the given sets are regions?
Answer
Sketches of the given sets are given in the following figure.
a)
b)
c)
d)
1. a,d
2. a
3. a
4. a, b, c
4