MATH 555 : PROBLEMS - 1
1. Prove the following identities :
(A) Im(iz) = Re(z) for any z ∈ C .
(B) Re(iz) = −Im(z) for any z ∈ C .
(C) |z|2 + a2 = |z + a|2 − 2Re(az) for any z ∈ C where a ∈ R .
(D) |z|2 + 2Re(bz) = |z + b̄|2 − |b|2 for any b, z ∈ C .
(E) |1 − āz|2 − |z − a|2 = (1 − |z|2 )(1 − |a|2 ) for any iff z, a ∈ C .
2. Compute (1 + i)n − (1 − i)n where n ∈ Z , n ≥ 0 .
3. Given c ∈ C with |c| = 1 and c 6= ±1 prove that
z−c =1
cz − 1
iff z ∈ R .
4. Given z, a ∈ C with |a| < 1, prove that
z−a < 1, = 1 , > 1
1 − āz
iff |z| < 1, = 1 , > 1 respectively.
5. Consider C with its standard structure as a vector space over the field R. The set {1, i}
constitutes the standard basis of C as vector space over R.
(A) Prove that each R-linear map ϕ : C −→ C is of the form
ϕ(z) = az + bz
for some a, b ∈ C.
(B) Compute tr(ϕ), det(ϕ) and ϕ−1 if det(ϕ) 6= 0 in terms of a, b ∈ C.
(C) Prove that an R-linear ϕ : C −→ C is C-linear iff ϕ ◦ J = J ◦ ϕ where J : C −→ C
is an R-linear map which is represented by the matrix
0 −1
1 0
with respect to the standard basis of C over R .
6. Let n ∈ N with n ≥ 2 and ω = eπ i/n .
(A) Show that
n−1
Y
xn − 1
=
x − ω 2k for every x 6= 1 .
x−1
k=1
(B) Show that
n−1
Y
1 − ω 2k = n .
k=1
(C) Show that ω
n(n−1)
2
= in−1 .
(D) Establish the Gauss Formula
n−1
Y
sin
kπ n
k=1
=
n
.
2n−1
(E) Employ the Gauss Formula to show that
Z π
log sin x dx = −π log 2
0
(F) Prove that
n−1
Y
k=1
cos
kπ n
=




0



(−1) 2
2n−1
if
n is even
.
n−1
if
n is odd
7. (A) Given A, B, C ∈ R, prove that the minimum and maximum values of the function
f (θ) = A + B cos 2θ + C sin 2θ
are A −
p
p
B 2 + C 2 and A + B 2 + C 2 .
(B) Prove that
|z|2 + |w|2 − |z 2 + w2 |
≤
2
| az + bw |2
a2 + b2
for any z, w ∈ C and a, b ∈ R with a2 + b2 6= 0.
1
Put cos θ =
a
b
and sin θ = 2
.
a2 + b2
a + b2
1
≤
|z|2 + |w|2 + |z 2 + w2 |
2