MATH 106 HOMEWORK 6 SOLUTIONS
1. (i) Give the Laurent expansion (in powers of z ), representing the function
1
f (z ) = 2
z 4z
in certain domains, and specify these domains. Moreover, for a counterclockwise simple
closed curve C inside each of the domains
HW7: Math 132 PDE II
Due on May 21st during class
In the following, test function is a smooth, fast-decaying function
(1) In HW6 problem 1 you computed the Fourier transform of
g(x) =
1 |x| |x| 1
0
otherwise
now use your result and Poisson summation formu
HW6: Math 132 PDE II
Due on May 14th during class
The Fourier transform in Rd is dened as f () =
Rd
f (x)eix dx.
(1) Let f, g be the functions dened by
f (x) =
1
0
if |x| 1
otherwise
and g(x) =
1 |x|
0
if |x| 1
otherwise
Compute the Fourier transform f ()
Math 106 - Midterm 1.
The exam consists of 6 questions. Each page of the exam is worth 10 points. The maximum
number of points is 50.
Name:
Acknowledgement and acceptance of honor code:
Signature:
Problem
Score
Problem 1
Problem 2
Problem 3
Problem 4
Prob
MATH 106 HOMEWORK 3 B SOLUTIONS
7. Solve the following equations
(i) ez = 1 + i 3,
(ii) e2z1 = i,
(iii) log z =
i
2.
Solution:
i
+ 2n, z = ln2 +
+ 2ni.
3
3
1
1 i
+ ni.
x = , y = + n, z = +
2
4
2
4
(i) ez = ex eiy = 2ei 3 x = ln 2 y =
(ii) e2z1 = e2x1 e2i
MATH 106 HOMEWORK 3 SOLUTIONS
1. Using the Cauchy-Riemann equations, show that if f and f are both holomorphic then f is a constant.
Solution: Let f = u + iv , so f = u iv . Since they are holomorphic, we can use the Cauchy-Riemann
equations:
ux = v y
uy
MATH 106 HOMEWORK 2 SOLUTIONS
1. Let z = x + iy and
f (z ) = x2 y 2 2y + i(2x 2xy )
Write f (z ) as a function of only z and z .
Solution: Using the formulas
x=
zz
z+z
and y =
,
2
2i
we compute
f (z ) = x2 y 2 2y + i(2x 2xy ) =
2
2
z+z
z+z zz
zz
zz
z+z
+i
Math 106 - Final Exam.
This is a closed book exam. No calculators are allowed. The exam consists of 8 questions worth
100 points. Good luck!
Name:
Acknowledgment and acceptance of honor code:
Signature:
Problem
Score
Problem 1
Problem 2
Problem 3
Problem
MATH 106 HOMEWORK 4 SOLUTIONS
1. Show directly from the denition that
sin(2z ) = 2 sin z cos z
Solution:
sin(2z ) =
(ezi ezi ) (ezi + ezi )
e2zi e2zi
=2
= 2 sin z cos z
2i
2i
2
2. Write the following complex numbers in standard form:
(i) (1 + i 3)i . What
MATH 106 HOMEWORK 5 SOLUTIONS
1. Let C1 denote the positively oriented circle |z | = 4 and C2 the positively oriented
boundary of the square whose sides lie along the lines x = 1, y = 1. With the aid of
Cor. 2 in Sec 46, point out why
f (z )dz =
C1
f (z )
Math 106, Practice exam.
Problem 1.
Let u(x, y ) = x2 y 2 2xy and v (x, y ) = x2 y 2 +2xy . Let f (x + iy ) = u(x, y )+ iv (x, y ).
Prove that f is entire. Compute f (i).
Problem 2.
Find the principal value of
e
(1 3i)
2
3i
.
Problem 3.
Let u(x, y ) = (ey
Math 106, Midterm 2.
Please return the exam before Friday, November 17, 4:30pm. This is an open-book exam, but
collaboration is not allowed.
Problem 1. [10 points]
Determine the principal values of the following complex numbers, and write them
in standard
HW4: Math 132 PDE II
Due on April 30th during class
(1) Find the Fourier cosine series of the function | sin x| in the interval (, ). Use it to nd
the sums
1
(1)n
and
.
4n2 1
4n2 1
n=1
n=1
(2) Solve the wave equation utt = c2 uxx for x (0, ) with boundary