Download Assignment 2

Math 892: Assignment 2 (due: February 25, 2016)
1. Let w ∈ C be a fixed complex number with |w| < 1. Let
z−w
.
f (z) =
1 − wz
Show that f is regular in |z| ≤ 1 and calculate f (w) and f 0 (w).
2. Let z1 , z2 , . . . , zn be distinct complex numbers. Let C be the circle around z1 such
that C and its interior do not contain zj for j > 1. Let
f (z) = (z − z1 )(z − z2 ) · · · (z − zn ).
Evaluate
Z
C
3. Find the radius of convergence of
∞
X
dz
.
f (z)
n!
n=1
zn
.
nn
4. Prove the following discrete version of integration by parts: For any two sequences of
numbers an , bn ,
SN :=
N
X
an bn = aN BN −
N
−1
X
Bn (an+1 − an ),
n=1
n=1
where
Bn =
n
X
bj .
j=1
Using this result, show that the power series
∞
X
zn
n=1
n
converges for every complex number z with |z| = 1, and z 6= 1.
5. Show that the series
∞
X
n=1
zn
n(n + 1)
converges for all |z| ≤ 1. What is its radius of convergence? Does the series
∞
X
n=1
converge for any z with |z| = 1?
1
zn
6. Let n be a non-negative integer. Define for α ∈ C, the binomial coefficient
α α(α − 1) · · · (α − n + 1)
=
.
n
n!
Show that the series
∞ X
α
n
n=0
zn
converges for |z| < 1.
7. Calculate the line integral
Z
L
dz
z
where L is the arc of the unit circle from 1 to i =
√
−1 traversing in the counterclock-
wise direction. More generally, what is the answer if i is replaced with eiθ for some θ
satisfying 0 ≤ θ ≤ 2π.
8. Compute
Z
sin z
dz
2
C z
where C is the circle of radius 1 centered at zero and oriented clockwise. What is the
answer if z 2 is replaced with z 3 in the integral?
9. Show that
2
max |ez | = e.
|z|≤1
10. If α, β, γ are the angles of a triangle and a, b, c are the lengths of the corresponding
opposite sides, show that
sin β
sin γ
1
sin α
=
=
= ,
a
b
c
d
where d is the diameter of the circumscribed circle of the triangle.
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