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DEPARTMENT OF MATHEMATICS AND STATISTICS
MATH. 421 – FINAL EXAM
12/16/99 NAME:
- (15 points) Given that the first few terms of the Laurent series for the function cot z
around z = 0 are:
cot z =
z
z
z
3
2 z
5
(i) Find the principal part at z = 0 of the function f (z) =
(1 + z) cot z
z
4
(ii) Find all the singularities of f (z) in the disk D = {|z| < 5 }. Determine the nature of
each singularity (isolated, removable, pole of what order, essential).
(iii) Find the residue at each isolated singularity in D.
- (10 points) Compute:
C
cos z
e
iz − 1
dz where C is the circle {|z| = 2} (traversed counter-
clockwise).
- (10 points) Compute:
C
(e
sin z
- ¯z)dz, where C is the circle {|z| = 2} (traversed
counter-clockwise).
- (15 points) Compute:
0
x
2
1 + x^6
dx
- (10 points) (a) Find the Laurent series of the function f (z) =
Log z
z − i
around the point
z 0 = i.
(b) Find the Taylor series of the function f (z) =
z^2 − 3 z + 2
around the point z 0 = 0.
- (5 points) Compute cos
π
− i ln 2
. Simplify your answer as much as possible.
- (5 points) Prove that
C
e
iz^2 dz
< 5 , where C is the piece of the circle |z| = 2 going
from 2 to 2i counter-clockwise.
- (5 points) Find an entire function f (z) such that Re(f ) = 4x
3 y − 4 xy
3 − y.
