1. HW1
(1) Evaluate the sum
sn (θ) = sin θ + sin 2θ + · · · + sin nθ
θ
for n ≥ 1 and θ ∈ R. (Hint: multiply sn by 2 sin and use cos A − cos B =
2
A+B
A−B
−2 sin
sin
)
2
2
(2) Let us denote
1
tn (θ) = + cos θ + cos 2θ + · · · + cos nθ, n ≥ 1,
2
and t0 (θ) = 1/2. For each m ≥ 1, define
σm (θ) =
t0 (θ) + t1 (θ) + · · · + tm (θ)
.
m+1
(a) Show that
"
#2
sin (m+1)θ
1
2
σm (θ) =
.
m+1
sin 2θ
(b) Compute
Z
1 π
σm (θ)dθ
π −π
The function σn (θ) is called the Fejer kernel.
(3) Given two integers n, m, let us denote a number δmn by
(
1 if n = m
δn,m =
0 otherwise.
Verify
the following facts:
Z 2π
(a)
cos nx cos mxdx = πδn,m ,
Z0 2π
(b)
sin nx sin mxdx = πδn,m ,
Z0 2π
(c)
sin nx cos mxdx = 0 for all n, m.
0
(4) For x ∈ R, we consider the functions:
−3
1
3
1
f (x) =
+ cos x − sin x + cos 2x + sin 2x,
2
2
4
4
and
1 1
2
−1
g(x) = + cos x − sin x +
cos 2x + 4 sin 2x.
2 2 Z
3
3 Z
Z 2π
2π
2π
Compute
f (x)2 dx, and
g(x)2 dx, and
f (x)g(x)dx.
0
0
0
(5) Let f (x) = cos µx for 0 ≤ x ≤ 2π where µ is not an integer. Compute the Fourier
expansion of f.
1