MATH 300 ASSIGNMENT 5-6: DUE MAR 4 (FRI) IN CLASS
The following exercises are on evaluating contour integral (from definition and
by using parametrizations):
A. Compute
R2
a) 0 cos(it)dt
R1
2
b) 0 (t3t+i)2 dt.
R
B. Evaluate the contour integral Γ (y + xi)dz over the contour Γ : z = t2 + it,
0 ≤ t ≤ 1.RHere x = Re(z), y = Im(z).
C. Compute Γ z̄dz over the semicircle z = eit , 0 ≤ t ≤ π.
The following are exercises on more recent lectures:
(1) Which of the following domains are simply connected?
a) the annulus 1 < |z| < 3
b) The set of all points in the plane except the set {z = x + yi : y =
0, −1 ≤ x ≤ 1}.
c) The interior of the ellipse x2 + 3y 2 = 1.
d) The unit disc with a slit D = {z = x + iy : x2 + y 2 < 1} \ {z = x + iy :
y = 0, x ≥ 0}.
(2) Let Γ be the circle |z| = 2 traversed counterclockwise once. Show that (try
to do the problem without looking at the hint below on page 2!)
Z
1
I=
dz = 0
2
Γ z(z − 1)
(3) Let C be the circle |z| = 2 traversed once counterclockwise. By using
cauchy’s integral formula, compute
Z
sin z
dz.
2
2
C (z + 1)
(4) Use Cauchy’s formula to show that if f is analytic inside an on the circle
|z − z0 | = r, then
Z 2π
1
f (z0 ) =
f (z0 + reiθ )dθ.
2π 0
Prove more generally that
Z 2π
n!
f (z0 + reiθ )e−inθ dθ.
f (n) (z0 ) =
2πrn 0
1
2
MATH 300 ASSIGNMENT 5-6: DUE MAR 4 (FRI) IN CLASS
Hint for problem 2:
(Hint: Use deformation invariance thm and consider for every R > 10,
Z
1
IR =
dz.
z(z
−
1)2
|z|=R
Show that I = IR and (by using M-L formula)
2π
|IR | ≤
.
2
(R − 1)