Complex Variables I
PROBLEM SET 7
Due October 31, 2006
1. Problem 4, page 181 of text.
2. Problem 7, page 189 of text.
1
3. Find the Taylor series of f = 2z about the point z0 = i. What is the radius R0 of the
largest disk with center at i where within wh
Complex Variables I
Final examination
December 19, 1006
ANSWER ALL QUESTIONS. JUSTIFY YOUR ANSWERS.
1. (25 points):
(a) Find the real and imaginary parts of 32i .
1 4 i
(b) What are the three values of (1 + i)1/3 , expressed in the form
rk eik , k = 1, 2,
o
ArJrcfw_ecr \l)
Rsrjrg,\)
?"o3F\lr
= t eTt =eZ"
r
f - r J i j- J q ' ' , o o
.
= cfw_,o5 i%,.\ln\q
e
h? E
,
O
), .
J#
=
cfw_*'
ii*-izni n r iScfw_ t
r
Lni n r "' at')?/'
"r4) = ail4iturni)=.-Wtzm).
cfw_ , t-* * + f > s L C*)'*fI' o * .
. 't. e >
e S (i
Complex Variables I
PROBLEM SET 10
Due November 28, 2006
This is the last homework set to be turned in for grading.
1. Problem 4., page 257 of text.
2. Problem 8, page 257 of text.
3.* Evaluate, using residue theory.
0
dx
.
(1 + x4 )2
(Hint: Use the conto
Complex Variables I
PROBLEM SET 5
Due October 17, 2006
1. Problem 2, page 153 of text. (Recall, if f is analytic within the region bounded by two
simple, closed, positively oriented contours, and is also analytic on the contours, then the integral
of f is
Complex Variables I
PROBLEM SET 8
Due November 7, 2006
1. Problem 3, page 213 of text.
2. Show that
z
=
nz n ,
(1 z )2 n=1
|z | < 1.
3. Problem 6, page 213 of text.
4. Problem 8, page 220 of text.
5. Show that the function
eNz ez
ez 1
f (z ) =
if z = 0
if
Complex Variables I
PROBLEM SET 3
Due October 3, 2006
1. (a) Find all values of z such that ez1 = 1 + i.
(b) Let the function f (z ) = u(x, y ) + iv (x, y ) be analytic in some domain D. State
why the functions U (x, y ) = eu(x,y) cos v (x, y ), V (x, y )
o
t c' ) l l \ t ,
ll-L
I
p\" r,rrr,-\t
u)
l r - 2 11 ' / -
=)
n*tr*L rl, .o^\r"d. tF i = t-
1
j
v o \n a /^e^hd r onT*"^\
z=t
oo
,
\
I
5\
/- ^ 1'. ( E*)*
\
rn:zO
Or' l
vr
? = t i oar
p"1", \
-*-L*-,)3
J
'r
o
e). rf
4"unka.un-
" -f,
4
t -?
b-, t u " I1 -V
Complex Variables I
PROBLEM SET 2
Due September 26, 2006
1. (a) Find limz i (z + 1/z ). (b) Show that limz z0 f (z )g (z ) = 0 if limz z0 f (z ) = 0
and there exists a positive number M such that |g | M in some -neighborhood of z0 .
2. Suppose that f (z0
Complex Variables I
PROBLEM SET 9
Due November 14, 2006
1. Using the Cauchy residue theorem evaluate the integral of each of the following functions
around the circle |z | = 3 taken in the positive sense.
(a)
exp(z 2 )
;
z5
(b)
exp(z )
;
(z 1)3
(c)
z+3
.
Complex Variables I
1. (a) Evaluate
(b) Verify that
PROBLEM SET 4
21
1t
izt
de
dt
iz
Due October 10, 2006
3
i dt.
= eizt , z = 0. Use this to evacuate
+1 izt
e dt.
1
2.
(a) Let w(t) be a continuous complex-valued function of t dened on the interval
a t b.
Complex Variables I
PROBLEM SET 6
Due October 24, 2006
1. Problem 2, page 163 of text.
2. If the points z = a and z = b are inside the domain bounded by a simple closed contour C ,
show that
e(za)(zb) dz
= 0.
C (z a)(z b)
3. We showed in class the the Leg
Complex Variables I
PROBLEM SET 1
Due September 19, 2006
1. (a) Find the modulus and the Arg of w where
w=
(4 2i)(1 + i)
.
(1 i)( 3 + i)
(b)Prove that, if z = x + iy is a complex number, |x| |z | |x| + |y |.
2. (a) Prove that, if z and w are complex numbe