M361 (56650) Midterm 2 Solutions
1. (a) (5 points) Let f (z ) = (1 cos z )5 . Find ord0 f .
Solution: We compute
ord0 f = 5 ord0 (1 cos z ) = 10
since 1 cos z = z 2 /2 + O(z 4 ).
(b) (5 points) Let g (z ) = sec z . Find resg /2.
Solution: We have
1
z /2
=
M361 (56650) Midterm 2 Solutions
1. (a) (5 points) Let f (z ) = (1 cos z )5 . Find ord0 f .
Solution: We compute
ord0 f = 5 ord0 (1 cos z ) = 10
since 1 cos z = z 2 /2 + O(z 4 ).
(b) (5 points) Let g (z ) = sec z . Find resg /2.
Solution: We have
1
z /2
=
Math361 HW # 8
Jonathan Campbell
Due: 4/14/2016
Problem 1 In class, I used the fact that
Lcfw_ty 0 (t)(s) = Y (s) sY 0 (s)
Prove this.
Problem 2 Show that
Lcfw_ty(t) =
d
Y (s)
ds
Problem 3 Let Hn (x) be the Hermite polynomials derived in class, i.e.
n
2
Solutions to Quiz 5 (M361)
1. (10 points) Show that the function f (z) = ez is nowhere analytic. (Hint: Remember that
there is no real number x such that both cos(x) = 0 and sin(x) = 0 at the same time.)
Proof 1. If z = x + iy, then
f (x + iy) = exiy = ex
Solutions to Quiz 8 (M361)
1. (5 points) Let C be the unit circle positively oriented. Evaluate the following contour
integral.
Z
cos(z)87 ez sin(z)120
dz
z2 + 4
C
Proof. The function
cos(z)87 ez sin(z)120
f (z) =
z2 + 4
is analytic away from z = 2i. Sinc
Solutions to Quiz 11 (M361)
1.
(1) (4 points) Find the residue
1
.
z(z 1)3
(2) (3 points) Let C be the positively oriented circle defined by the equation
|z 1| = 12 . Evaluate the integral
Z
dz
.
3
C z(z 1)
Res
z=1
Proof.
(1) If we choose (z) = z1 , then
Solutions to Quiz 10 (M361)
1. Let
f (z) =
1
.
z(z 1)
Moreover, let C1 be the circle |z| = 12 , C2 the circle |z 1| = 21 , and C3 be the circle |z| = 2
all positively oriented.
(1) (3 points) Find the Laurent expansion of f (z) around z0 = 0.
(2) (1.5 poi
Solutions to Quiz 13 (M361)
1.
(1) (3 points) What are the singularities of
1
?
+8
Which of them lie in the upper half plane?
(2) (3 points) Find the following residue for all singularities z0 of f in the upper half
plane:
e3iz
Res 2
.
z=z0 2z + 8
(3) (3
Solutions to Quiz 12 (M361)
1. Let f (z) = sin(z)2 sin(z).
(1) (4 points) What is the order of the zero z = 0 of f (z)?
(2) (3 points) Determine the residue
1
Res
.
z=0 f (z)
Proof.
(1) We have f (0) = sin(0)2 sin(0) = 02 0 = 0. Moreover,
f 0 (z) = 2 sin(
Solutions to Quiz 4 (M361)
1. (15 points) For which complex numbers z is the function f (z) = z 2 z z
differentiable? Compute the derivative of f at the values at which it exists.
Proof. Let z = x + iy. Then
f (x + iy) = (x iy)2 (x iy) (x + iy) = x2 y 2 2
Solutions to Quiz 6 (M361)
1. (7 points) Evaluate the following integral. Give your answer in the form a + ib for real
numbers a and b.
Z
sin(it)dt
0
Proof. We have
Z
0
1
sin(it)dt = cos(it)
= [i cos(it)]t=0
i
t=0
e + e
= i cos(i) i cos(0) = i
i
2
e +
Solutions to Quiz 7 (M361)
1. (8 points) Let C be the circle given by z(t) = 2ei , where 0 2. Evaluate
Z
dz
.
2
C (z i)
Proof. The function
f (z) =
1
(z i)2
has the antiderivative
1
.
(z i)
Since additionally C is a closed contour, we get
Z
dz
= 0.
2
C (z
Solutions to Quiz 3 (M361)
1. (8 points) Use the definition of the derivative to give a direct proof of the fact
d 2
(z z) = 2z 1.
dz
Proof. We have
d 2
(z + (z)2 (z + (z) (z 2 z)
(z z) = lim
(z)0
dz
(z)
2
z + 2z(z) + (z)2 z (z) z 2 + z
= lim
(z)0
(z)
2
2
Math361 HW # 9
Jonathan Campbell
Due: 4/28/2016
Problem 1 Prove that
1
a
Z
a/2
e2inx/a e2imx/a dx = mn .
a
2
We used this fact in the construction of Fourier series.
Problem 2 Compute the fourier transform of cos x2 (Hint: youre going to have to
use the G
Math361 HW # 7
Jonathan Campbell
Due: 3/31/2016
Problem 1 Find an expression for the sum of the series
X
n2 z n
|z| < 1.
n=1
Hint: it is the derivative (or perhaps second derivative of a certain series).
Justify convergence.
Problem 2 Use the Taylor serie
Math361 HW # 6
Jonathan Campbell
Due: 3/10/2016
Problem 1 Show
2
Z
2
d
=
a + b sin
a2 b2
0
Problem 2 Show that
Z
Problem 3 Compute
Z
0
Problem 4 Evaluate
Z
0
Problem 5 Prove that
2
Z
0
Problem 6 Evaluate
1
2i
d
= 2
2
1 + sin
Z
C
(x2
dx
+ 1)(x2 + 4)2
cos
Math361 HW # 3
Jonathan Campbell
Due: 2/11/2016
Problem 1 Show that
1
f (z) =
2
1
z+
z
is a conformal map from the upper half disc (D H) to the upper half
plane.
Problem 2 Use the previous exercise to compute where sin z takes the strip
S = cfw_z : Im(z)
Math 361 HW 5
Due: March 3, 2016
Problem 1 Compute
Z
sin 5z
dz
z + /2
where is cfw_|z| = 5.
Problem 2 Compute
eiz
dz
z3
Z
where = cfw_|z| = 2.
Problem 3 Compute
1
2i
Z
z 2 dz
z2 + 4
where is a square with vertices at 2 and 2 + 4i.
Problem 4 Show that
Z
2
Math361 HW # 4
Jonathan Campbell
Due: 2/18/2016
Problem 1 Let : [a, b] be a curve. Show that (t) = (b t) + a) is that same
curve but parameterized backwards. For f : C a complex function,
show
Z
Z
f (z) dz = f (z) dz
Problem 2 Show that if |a| < r < |b| t
Math361 HW # 2
Jonathan Campbell
Due: 2/04/2016
Problem 1 Show that a triangle with vertices a, b, c is equilateral if
a + b + c 2 = 0
where is the 3rd root of unit e2i/3 .
Problem 2 Recall that the centroid of a triangle is the intersection of the lines
Math361 HW # 1
Jonathan Campbell
Due: 1/28/2016
Problem 1 Heres some practice computing with complex numbers.
1. Express, with no complex denominator,
2
5 12i
5 + 5i
2 + 3i
1i
1+i
2. Write ( 3 + i)78 as a complex number in the form x + iy.
3. Convert 10ei
Statistics 100A
Homework 4 Solutions
Ryan Rosario
Chapter 4
39. A ball is drawn from an urn containing 3 white and 3 black balls. After the ball is drawn, it
is then replaced and another ball is drawn. This goes on indenitely. What is the probability
that
American
Regionalism
Halley Cortez, Nicole Hanlon, Glenn Cable
History
Emerged in the Midwest in
the early 1930s
Rejection of modernism
label and foreign influence
Depicting rural lifestyle as
the economy failed
As the U.S. increased its
isolation, so did
f Art
ue o
Val
Th e
Hanlon
Nicole
Artworks
Result of who makes
them as well as
influenced by the input
of others!
Others
Patrons who employ an
artist to make a work
Collectors who buy
Dealer and gallery
owners who sell
Japanese Culture
Some art is not mea
Math 362K Probability
Fall 2007
Instructor: Geir Helleloid
Practice Midterm 1: Solutions
1. In a group of 3120 entering freshmen, 3002 have 2 years of algebra, 2962 have two
years of English, and 2340 have two years of languages. If 2902 have both algebra
Solutions to Quiz 9 (M361)
1. (8 points) Find the Laurent series that represents the function
ez
f (z) =
z
in the domain 0 < |z|.
Proof. We have
X
ez
zn
1
=z
z
n!
n=0
=
X
z n1
n=0
n!
zm
1 X
.
= +
z m=0 (m + 1)!
(That last variable change m = n 1 is not ne