M519 Sample Final
All problems would be weighted equally, 10 Problems would be graded.
♯ 1: Find the Laurent series for f (z), |z| > R, R > 0, where
Z ∞ −1/t
e
f (z) =
dt.
t2
z
Classify the singularity at z = 0 and determine the residue.
♯ 2: Let P (z) = a0 + a1 z + a2 z 2 + · · · + an z n be a polynomial that satisfies |P (z)| 6 M
for |z| = 1. Show that each coefficient ak satisfies |ak | 6 M .
♯ 3: Classify the singular points of f (z) in the extended complex plane and give the
residue for all isolated singularities. Give the order of the pole in the case of a pole.
(a) f (z) =
ez −1
.
z3
(b) f (z) =
z(z−π)2
.
(sin z)2
♯ 4: Find the Laurent series in the annulus 1 < |z| < 2 for
f (z) =
♯ 5: Evaluate
R∞
x sin(x)
dx.
−∞ 4+x2
1
.
(z − 1)2 (z − 2)
Justify the use of any theorems from class.
R∞ √
♯ 6: Evaluate 0 1+xx2 dx by using an appropriate “keyhole contour”. Justify the use of
any theorems from class.
♯ 7: Evaluate
R π/2
0
dθ
.
1+cos2 θ
♯ 8: Show that the equation 2z 5 + 8z − 1 = 0 has
(a) no roots in |z| > 2,
(b) one root in |z| < 1 and that this root is real and positive,
(c) exactly four roots in the annulus 1 < |z| < 2.
♯ 9: Let P (z) = an z n + · · · + a1 z + a0 with an , a0 6= 0, and assume P (z) 6= 0 on |z| = 1.
Let P ∗ (z) = a0 z n + a1 z n−1 + · · · + an . Prove that if |a0 /an | > 1, then P (z) has the same
number of roots in |z| < 1 as does the polynomial a0 P (z) − an P ∗ (z).
♯ 10: Consider the birational transformation w = f (z) ≡
following regions under f (z)?
(a) {reiθ | 0 6 r 6 1, 0 6 θ 6 π}.
(b) {reiθ | 0 6 r 6 1, π 6 θ 6 2π}.
z−i
.
z+i
What is the image of the
♯ 11: Find a bilinear transformation that maps the half–plane above the line y = x onto
the interior of the circle |w − 1| = 1.