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CAN702S - COMPLEX ANALYSIS - 1ST OPP - NOV 2022 |
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nAm I BIA un IVE RSITY
OF SCIEnCE
TECHnOLOGY
FACULTY OF HEALTH, APPLIED SCIENCES AND NATURAL RESOURCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of Science in Applied Mathematics and Statistics
QUALIFICATION CODE: 07BAMS
LEVEL: 7
COURSE CODE: CAN702S
COURSE NAME: COMPLEX ANALYSIS
SESSION:
DURATION:
NOVEMBER 2022
3 HOURS
PAPER:THEORY
MARKS: 100
EXAMINER
FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
DR. NEGACHERE
MODERATOR:
PROF. FORTUNE' MASSAMBA
INSTRUCTIONS
1. Answer ALL the questions in the booklet provided.
2. Show clearly all the steps used in the calculations.
3. All written work must be done in blue or black ink and sketches must
be done in pencil.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPER CONSISTS OF 3 PAGES (Including this front page)
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QUESTION 1 [19]
1.1. Express z = Hi in the form of a+ ib and then find its modules.
[5]
3+i
1.2. Use exponential form to express (1 - i) 98 in the form of a+ i b.
[9]
1;. 1.3. Find the argument and the principal argument of z = - - i
[5]
QUESTION 2 [8]
Findthe image of the triangle with vertices z1 = -2 + i and z2 = 2 + 2i and z3 = -2 + i
under the mapping w = f(z) = (2 + i)z - 2i. Skitch the triangles.
QUESTION 3 [24]
3.1. Verifythe Cauchy-Riemann equations for f(z) = iz2 + z.
[10]
3.2. Show that f(z) = e-z is analytic using the Cauchy-Riemann equations.
[14]
QUESTION 4 [16]
= Verifythat u(x, y) x3 - 3xy 2 + 3x 2 - 3y 2 is harmonic, and find its harmonic conjugate
v(x, y}. If f(z}= u(x, y}+ i(x,y},with f(O}= i , find f(z}.
QUESTION 5 [15]
Compute the following integrals.
JJ1t
5.1. teit dt.
[7]
= 5.2. fc(z 2 - z2)dz where C:z(t) t 2 + it, 0 :£ t :£ 1.
[8]
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QUESTION 6 [18]
Evaluate the following integrals
Jc 6.1. ( sin;z dz where C is the positively oriented contour shown in the figure below. [11]
z +1
y
f = 6.2.
1
z3 ( z+31.)
dz
where
C is the circle
lz -
ii
1 is oriented positively.
[7]
C
END OF FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
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