ASSIGNMENT 1 SOLUTIONS
MAT 572 A FALL 2007
Problem 1 (cf. [1, Exercise I.4.5]). Let z = cis(2/n) for an integer n 2. Show that
1 + z + z 2 + + z n1 = 0.
Proof. Let s denote the value of the sum. Since z n = 1, we have
zs = z + z 2 + + z n1 + z n = z + z 2
ASSIGNMENT 2 REVISED SOLUTIONS
MAT 572 A FALL 2007
Problem 1 (cf. [1, Exercise III.2.9]). Prove that the principal branch of the logarithm is
continuous.
Proof. Let G = C \ cfw_z | z = (z ) 0, and suppose rn exp(in ) r exp(i) in G with rn
and r positive a
ASSIGNMENT 3 SOLUTIONS
MAT 572 A FALL 2007
Problem 1 (cf. [1, Exercise III.2.13]). Let G = C \ cfw_z | z = Re z 0, and let n be a
positive integer. Find all analytic functions f : G C such that z = (f (z )n for all z G.
Solution. For each positive integer
ASSIGNMENT 4 SOLUTIONS
MAT 572 A FALL 2007
Problem 1 (cf. [1, Exercise III.3.10]). Find all Mbius transformations which map the disc
o
D = cfw_z | |z | < 1 onto itself.
Solution. First, any such Mbius transformation T must also take Dc onto Dc , since Mbi
ASSIGNMENT 5 SOLUTIONS
MAT 572 A FALL 2007
Problem 1 (cf. [1, Exercise IV.1 #7]). Show that : [0, 1] C dened by
(t) =
t + it sin(1/t) t = 0
0
t=0
is a path, but is not rectiable.
Proof. Since limt0+ t sin(1/t) = 0, is continuous on [0, 1], so is a path.
ASSIGNMENT 6 SOLUTIONS
MAT 572 A FALL 2007
Problem 1 (cf. [1, Exercise IV.2 #2]). Prove that if G C is open and : I G is a
rectiable curve, and : (I ) G C is continuous and g : G C is dened by
g (z ) =
(w, z ) dw,
then g is continuous. Also prove that if
ASSIGNMENT 7 SOLUTIONS
MAT 572 A FALL 2007
Problem 1 (cf. [1, Exercise III.3 #17]). Show that if G is open and connected, f : G C
is analytic, and f (G) is a subset of a circle, then f is constant.
Proof. Let be a circle in C with f (G) . Since G is conne