MATH 321 - HOMEWORK #10
P ROBLEM 1. Consider the function f(x) on [−π, π]:
0 if x ∈ [−π, 0]
f(x) =
x if x ∈ (0, π]
(a) Find the Fourier series
f(x) ∼
X
cn einx .
n
(b) Express the Fourier series in terms of sines and cosines.
(c) Evaluate the Fourier series at x = 0 and x = π/2. Assuming that the series converges
to f(x) at those points, find formulas for π and π2 in terms of series of rational numbers.
P ROBLEM 2. Consider Fourier series
f(x) ∼
f(x) ∼ a0 +
X
X
cn einx ,
n
an cos(nx) + bn sin(nx).
n>0
Prove that
(a) If f(x) is even, f(−x) = f(x), then bn = 0.
(b) If f(x) is odd, f(−x) = −f(x), then an = 0.
(c) If f(x) is real valued, then c̄n = c−n .
(d) If f(x) satisfies f(−x) = f(x), then cn are real.
P ROBLEM 3. For f(x) and g(x) 2π-periodic functions on R, define their convolution
Z a+2π
f ∗ g(x) =
f(x − y)g(y)dy,
a
^
where a can be taken any real number. Let f(n)
denote the n-th coefficient cn in the Fourier
series of f(x). Prove that
^ g(n).
(fd
∗ g)(n) = 2πf(n)^
In other words, the Fourier coefficients of the convolution are the products of Fourier
coefficients of the factors.
P ROBLEM 4. Fourier series can be used to find particular solutions to differential equations.
Consider the differential equation for the unknown function y(x):
y 00 + 2y 0 + y = sin(x).
Find a solution y(x) in the form of a Fourier series y(x) ∼ a0 +
1
P
n>0
an cos(nx)+bn sin(nx).