Math 301
Test # 1 Solutions
1. All contours are in the positive direction.
(a)
I
1
e1/z cos( )dz ; C : |z | = 1
I=
2z
C
Here f (z ) = e1/z cos( 21z ) and z = 0 is an essential singularity, so the residue has to be
calculated directly.
1
1
1
1
) = (1 + + 2
2014-02-12
Math 301 Midterm 1 solutions
10
1. Evaluate I =
(x2
Page 1 of 3
1
dx.
+ 4)2
Solution:
We use a semi-circle in the upper half plane. As shown in class, since the denominator has
degree 4, the integral along the arc tends to 0 as R , and thus I =
MATH 301, sample midterm 1
This is much longer then the actual midterm, and intended only to give
you an idea what kind of questions to expect.
1. Classify the singularities of
of them.
e1/z
sin z
and nd the residue at each
Essential at 0 and the residue
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Math 301 Final Exam Apr 24, 2008 Duration: 150 minutes Name: Student Number:
1. Do not open this test until instructed to do so! 2. Please place your student ID (or another picture ID) on the desk. 3. This exam should have 4 pages, including this cover sh
Math 301, Test 4 Solutions
.
1. Solve:
2 = 0 in cfw_Re z > 0 cfw_|z 5| > 4
with :
= 4 on Re z = 0 and = 2 on |z 5| = 4
y
4
2
-5
5
-2
10
x
-4
Solution: map the region onto concentric circles in the w-plane.
Map inverse points z = 3 into w = 0 and w = . He
Math 301 Solutions to Test 3
1. (a) Consider the function f (z ) = z 3 4z 2 + 2z 1
On the imaginary axis,
f (iy ) = 4y 2 1 + i(y 3 + 2y )
= (2y + 1)(2y 1) iy (y
2)(y + 2).
Re f (iy ) = 0 when y = 1 . Im f (iy ) = 0 when y = 0, 2.
2
There are no common ro
Math 301
Test # 2 Solutions
1. Evaluate the integral
I=
Z
0
Log (x)
dx
4 + x2
Solution: take branch cut along either the positive or negative real axis; integrate around semi-cirlce contour in the uper half plane containing a singularity
of f (z ) = Log (
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2014-03-19
12
Math 301 Midterm 2
Page 1 of 2
1. Let f is a bilinear map with pole at 1 and f (i) = 1, f (1) = 0.
a. Find f ?
Solution: The values at 1 and 1 imply f (z) =
a = i, so f (z) = i(z+1)
z1
a(z+1)
.
z1
To get f (i) = 1 we have
b. What is the imag