MATH 5321
Exam I
Make-up
18 February 2009
Answer the problems on separate paper. You do not need to rewrite the problem statements on
your answer sheets. Work carefully. Do your own work. Show all relevant supporting steps!
1.
X
2.
X
3. (28 pts)
Identify each of the finite isolated singularities for each of the following
functions. Determine whether the singularities are removable singularities, poles
or essential singularities. If the singularity is removable, determine what value
the function should be assigned at the singularity to analytically extend the
function at the singularity. If the singularity is a pole, determine the order of the
pole. If the isolated singularity occurs at z = 0, then determine residue of the
singularity.
cos( π2z )
a. (10) f ( z ) = 2
z ( z + 1) 2
ez −1
f ( z) = 3
z ( z + 1)
b. (6)
d. (6)
1⎞
⎛1
f ( z ) = ( z 2 − z 4 ) ⎜ − sin ⎟
z⎠
⎝z
2
c. (6)
4. (20 pts)
sin( π2z ) 1 − z
z2 − z
f ( z) =
Prove the following corollary to Cauchy’s Theorem (Version #4):
Corollary. Let G be a region in which is simply connected and let
f ∈ A (G ) . If f ( z ) ≠ 0 on G, then f has a branch-of-square-root f on G,
i.e., there exists a g ∈ A (G ) such that g 2 = f .
5. (21 pts)
Classify each of the following regions at to whether they are convex or not
convex, starlike (with respect to some point a ∈ G ) or not starlike (with respect
to any point a ∈ G ) , simply connected or not simply connected.
Notation: [a,b] denotes the (closed) straight line segment between a and b.
Q1 denotes the first quadrant.
Use the table on the next page to report your answers
N.B. Mark each cell in the table as Y or N, i.e., do not leave any cell in the table
blank.
a.
Ga = B (0,1) \[0,1)
c.
Gc = B (0,1) \ {∪ [ei 2π / n , nn−1 ei 2π / n ]} d.
b.
Gb = B (0,1) \ B ( 1 2 , 1 2 )
∞
n=2
Gd = B (0,1) ∪ B (1,1)
∞
e.
Ge = B (0,1) \ {∪ [ei 2π / n , nn−1 ei 2π / n ]}
f.
G f = Q1 \ B (4 + i , 3)
n =1
Gh = B (1,1) ∪ B (e i 2π / 3 ,1) ∪ B (e − i 2π / 3 ,1)
g.
G g = Q1 \ B (4 − 2i, 3)
i.
Gi = H i ∪ {B ( −1, 1 2 ) ∪ B (1, 1 2 )} where H i = { z : | Re z | < 1}
j.
G j = H j ∪ {B ( −1,1) ∪ B (1,1)} where H j = { z : | Re z | < 1 and Im z > 0}
k.
Gh = B (1, 1110) ∪ B (i, 10 10 ) ∪ B ( −1, 10 10) ∪ B ( −i, 10 10 )
l.
Gl = Q1 \ {∪ [1 / n,1 / n + i / n ]}
m.
Gm = LHP \ { z : z = 2e it ,| t |< 1.6} where LHP = { z : Re z < 0}
n.
Gn = Q1 \ {[3i , 3i + 3] ∪ [6, 6 + i ]}
h.
∞
n =1
Name ___________________________
Convex
Starlike
(Return this sheet with your other problems)
Simply
Connected
Convex
Ga
Gh
Gb
Gi
Gc
Gj
Gd
Gk
Ge
Gl
Gf
Gm
Gg
Gn
Starlike
Simply
Connected