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MATH 621 – Spring 2005 Homework Set # 4
- a) Describe the curve C parametrized by γ(t) = a cos t + i b sin t,
t ∈ [0, 2 π]. Compute
C
dz z
b) Compute
∫ (^2) π
0
dt a^2 cos^2 t + b^2 sin^2 t
- Let γ : [0, 1] → C be a C^1 -curve. Define:
f (z) =
γ
ξ − z
dξ
a) Prove that f is holomorphic in C \ γ([0, 1]). b) Show that if γ(t) = t, it is not possible to extend f to a continuous function in all of C.
- Let f be an entire function and suppose there exist positive inte- gers k and m such that
(1 + |z|k)−^1 ·
dmf dzm
is bounded. Prove that f is a polynomial and estimate deg(f ) in terms of k and m.
- a) Determine all entire functions f such that
f (1/n) = f (− 1 /n) = 1/n^3
for all n ∈ N.
b) Determine all entire functions f such that
f ′′(1/n) + f (1/n) = 0
for all n ∈ N.
c) Determine all entire functions f such that
n−^3 /^2 ≤ |f (1/n)| ≤ 2 n−^3 /^2
for all n ∈ N.
- Show that: ∫ (^2) π
0
e^2 t^ cos^ θ^ dθ = 2π
∑^ ∞
n=
tn n!
Hint: Let z = eiθ, then 2 cos θ = z + z−^1.
2
- Let f be holomorphic in an open set U containing the disk D = {|z − z 0 | ≤ b}. Show that ∫ ∫
D
f (x + iy) dy dx = πb^2 f (z 0 )
Hint: Use polar coordinates and Cauchy’s formula.
- Suppose f (z) =
∑^ ∞
n=
an(z − z 0 )n^ for |z − z 0 | < R. Show that:
2 π
∫ (^2) π
0
|f (z 0 + reiθ)|^2 dθ =
∑^ ∞
n=
|an|^2 r^2 n^ ; 0 < r < R
- Let f (z) = cos
( (^) 1 + z 1 − z
, |z| < 1. Find the set Zf := {z ∈ C :
f (z) = 0}. Does Zf have any accumulation points? Explain why this does not contradict the result proved in class.
- Find the Taylor expansion of each of the following functions and determine its radius of convergence:
a) f (z) = sin^2 z ;
b) f (z) =
2 z + 1 (z^2 + 1)(z + 1)
c) f (z) =
ez 1 − λz
, λ ∈ C.
d) Suppose that the function in (b) is expanded in a power series around z = 1. What would be the radius of convergence of that series?
- Evaluate: (^) ∫
Γ
cosn(z) z^3
dz ,
where n is a non-negative integer, and Γ is the unit circle {|z| = 1} traversed counterclockwise once.
