Be sure this exam has 9 pages including the cover
The University of British Columbia
MATH 305
Final Exam – Dec 6, 2013
Family Name
Given Name
Student Number
Signature
This exam consists of 3 parts. No notes nor calculators. Note the number of marks for each question. Use
your time wisely. Time: 2 21 hours
Problem
max score
1.
8
2.
28
3.
14
total
50
score
Dec 6, 2013
(8 points)
(2 points)
Math 305 Final
Page 2 of 9
1. Answer the following multiple choice questions. Check your answer very carefully. Your answer will be
marked right or wrong (work will not be considered for this problem).
(a) Let f (z) =
A.
B.
C.
D.
E.
1
z 2 (1−z)3 .
Then the residue of f at z = 1, denoted as Res(f ; 1) is
6
−6
3
−3
None of the above.
(a)
(2 points)
2
2
(b) Let f (z)
R = 1/(z + 1). Let Γ be the positively oriented eclipse {z = x + iy : x +
Then Γ f (z)dz =
A. 0
B. π
C. −π
D. π/2
E. None of the above.
(y−2)2
4
= 1}.
(b)
(2 points)
(c) Which of
A.
B.
C.
D.
(2 points)
P∞
the following is/are true? (There may be more than one choices!)
The function f (z) = sin z/ cos z has only isolated singularities which are simple poles
z
1
The function f (z) = cos
z 2 − z 2 is bounded in the disk {z : |z| ≤ 1}
There does not exist a bounded function which is analytic in the region {z : |z| > 1}
None of the above hold true.
(c)
(d)
1
k=1 k2 +4
A.
B.
C.
D.
=
(hint: use Residue Theory)
(2π)
π cosh
sinh(2π)
sinh(2π)
π cosh(2π)
π sinh(2π)
4 cosh(2π)
π cosh (2π)
4 sinh(2π)
E. None of the above
(d)
Dec 6, 2013
Math 305 Final
Page 3 of 9
(28 points)
2. Compute the following integrals and simplify your answer to the best possible. Show in detail how you
arrive at your answer (work will be considered for this problem, DO NOT omit steps!).
(8 points)
(a) Find the Laurent series for f (z) = (z−1)12 (z−3) valid for (A) 0 < |z − 1| < 2 and (B) 0 < |z − 3| < 2.
You need to write out the explicit expression for the first four terms.
Dec 6, 2013
(8 points)
Math 305 Final
(b) By using the Residue Theory, compute the integral
Page 4 of 9
R∞
x cos x
dx
−∞ x2 −3x+2
Dec 6, 2013
(5 points)
Math 305 Final
Page 5 of 9
(c) By using the Residue Theory, compute the integral
Z ∞
x
I=
dx
3
x +1
0
(c)
Dec 6, 2013
(7 points)
(d) The value of the integral
Math 305 Final
Rπ
cos(x−i)
dx
0 sin(x+i)
Page 6 of 9
is
(d)
Dec 6, 2013
(14 points)
(7 points)
Math 305 Final
Page 7 of 9
3. By using the Residue Theory, compute the following integrals. You must explain clearly which
theorems you are using and justify all your estimates.
(a) Compute
Z
I = P.V.
0
∞
(1 − cos x)2
dx.
x4
Dec 6, 2013
(7 points)
Math 305 Final
(b) Let 0 < λ < 1. Compute
Z
I=
0
∞
xλ (x
1
dx.
+ 1)(x + 4)
Page 8 of 9
Dec 6, 2013
This is an extra page for answers.
Math 305 Final
Page 9 of 9