KWAME NKRUMAH UNIVERSITY OF SCIENCE & TECHNOLOGY, KUMASI
COLLEGE OF SCIENCE
DEPARTMENT OF MATHEMATICS
BSc Mathematics, Third Year
Third Year First Semester Examination, 2023
Mid Semester Examination, February 2023
MATH 361: COMPLEX ANALYSIS
February 27, 2023
Time: 1 : 00 Hour
INDEX NUMBER:
INSTRUCTIONS
1. You are advised to peruse all instruction before commencing this paper.
2. . In your own interest you are advised to write down your index number in the space
provided on this page and do all rough work and calculations in the answer booklet
provided.
3. Candidates are strongly advised to observe all COVID-19 protocols before, during and
after the examination.
4. Your responsibility in the examination is to answer all the 20 questions. Each question
has five choices, A – E. You are to indicate precisely and clearly your choice by checking
or circling the LETTER corresponding to your choice in the question booklet, and again
to shade the correct choice on the scannable form provided.
5. Candidates are to note that all symbols used in this examination have their usual meanings.
6. You are advised to submit the answer booklet, the question booklet and the scannable
form to the invigilator immediately the examination ends.
7. Each question in this examination carries 1 mark and the exam has a total score of 20
marks.
K. Piesie
Page 1 of 6
1. Find the value of (−i)i .
A. exp( π2 )
B. e−π
C. 1
D. exp( π2 − 2nπ)
E. None of the choices above
√
6
3i)
2. Find the modulus of the complex number z = (1−
.
(1+i)4
A. 4
B. 16
C. -16
√
D. 4 2
E. None of the above choices .
3. Simplify in when n even.
A. πi
B. 2π
C. ±1
D. ±i
E. None of the above choices .
4. Let f (z) = (z−1)23z−2
. Which of the choices below is/are not a simple pole(s)
(z+1)(z−4)
of f (z)?
I: −4
II: −1
III: 4
A. I and II only
B. I only
C. III only
D. II and III only
E. None of the above choices A, B, C or D.
K. Piesie
Page 2 of 6
5. Which of the following fully describes the locus of {z : Arg(z) = π6 }
A. y = 3x
B. y = √13 x
C. y =
√
3x
√
D. y = − 33 x
E. None of the above choices .
6. Which region in the complex plane is described by the equation | z − 4i | +
| z + 4i |= 0?
A. A circle centred at z = −4 + 4i with radius 42
B. A concentric circle centered at the origin.
C. An ellipse whose foci are (0, ±4)
D. A hyperbola with foci on the real axis.
E. None of the above choices .
7. Which of these functions is not harmonic?
I: f (x, y) = sin x sinh y
II: f (x, y) = cosh 2x sinh 2y
III: f (x, y) = 3xy 2 −x3 +2x
A. I, II and III only
B. I only
C. II only
D. I and II only
E. None of the above choices A, B, C or D.
8. Let C be a circle of radius r centred at z0 with parametric representation
Z
θi
z = z0 + re , 0 ≤ θ ≤ 2π. Evaluate
|z − z0 | dz.
C
A. r2 e2πi
B. rθ
C. 2πr
D. 21 r2 θ
E. None of the above choices.
K. Piesie
Page 3 of 6
9. Which of the following is a real-valued function of the complex variable z ?
I: f (z) = ezi
II: f (z) = |z|
′
III: f (z) = |zz ′ |
A. I only
B. II and III only
C. III only
D. II only
E. None of the above choices A, B, C or D.
10. Let z be a nonzero complex number, ln z a complex-valued function and Ln z
the principal part of ln z. Find Ln(−ei).
A. 1 − 23 πi
B. 1 − 12 πi
C. 12 πi
D. − 32 πi
E. None of the above choices.
11. Solve the equation | eθi − 1 |= 2 for all values of θ for which 0 ≤ θ ≤ 2π.
A. π
B. eπi
C. 2π.
D. eπ
E. None of the above choices.
12. Find all values of z such that ez = −1.
A. z = 2nπi
B. z = (2n + 1)π
C. z = (2n + 1)πi
D. 2nπ
E. None of the choices above
K. Piesie
Page 4 of 6
Z
13. Evaluate
f (z)dz where f (z) =
√
z and z = 3eθi , 0 ≤ θ ≤ π.
√
A. A. 2 3(1 + i)
√
B. 2 3(1 − i)
√
C. −2 3(1 + i)
√
D. −2 3(1 − i)
E. None of the above choices.
14. Find the value of Ln z when z = 1 − i.
A. 1 − 21 πi
B. 1 + 32 πi
C. 14 (ln 4 − πi)
D. 12 ln 2 − 14 πi
E. None of the above choices .
15. Which of these equations do/does not satisfy the conjugate root theorem?
I: z 2 + 2iz − 3 = 0
II: z 3 + (iz)2 − 3z − 3 = 0
III: z 2 + (6 − 4i)z − 16 − 2i = 0
A. I only
B. II only
C. II and III only
D. I and II only
E. None of the above choices A, B, C or D.
16. Which of these functions satisfy/satisfies the Cauchy-Riemann equations?
I: f (z) = z 2
II: f (z) = |z|2
III: f (z) = ez
A. I and II only
B. III only
C. II only
D. I and III only
E. None of the above choices A, B, C or D.
K. Piesie
Page 5 of 6
17. If f (x, y) = y 3 − 3x2 y is harmonic throughout the entire plane. Find a harmonic
conjugate for f .
A. −6xy + c
B. −3x2 y + x + c
C. −3x2 y + x2 + c
D. x3 − 3xy 2 + c
E. None of the above.
Preamble: Let x ∈ R and define f (x) by the formula f (x) = e(1+i)x . Use this
preamble to answer question 18.
Z π
f (x)dx.
18. Find Re
0
A. − 12 (1 + eπ )
B. − 12 (1 − eπ )
C. − 12 (1 + e−π )
D. − 12 (1 − e−π )
E. None of the above choices.
Z
iθ
19. If z = 2e , −π ≤ θ ≤ π, evaluate
z̄dz.
A. 8π
B. 4πi
C. 4π
D. 2πi
E. None of the above choices .
20. If z z̄ =| z |2 , deduce from question 19 above the value of
Z π
1
dz.
z
−π
A. 14 π
B. 14 πi
C. 2πi
D. π
E. None of the above choices .
K. Piesie
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