Math 257 – Assignment 5. Due: Wednesday, February 9
1. Orthogonality. Let f (x) = 1 + cos 3πx + cos 5πx + cos 6πx + sin πx + sin 7πx. Compute
2
R1
f
(x)
dx.
−1
(Hint: remember how we used the identities for the integrals of multiples of sine and cosine
function. For these identities, see Problem 3 in HW 4.)
2. Fourier series. Let f (x) be the 2-periodic function defined by
(
λ −1 ≤ x < 0
f (x) =
( where λ > µ are positive constants.)
µ 0≤x<1
(a) Graph the function f (x), −∞ < x < ∞.
(b) Find the Fourier series of f (x).


 λ − µ −1 ≤ x < 0
2
(c) Find the Fourier series of the 2-periodic function g(x) = µ −
λ


0≤x<1
2
(Hint: this problem (c) is very easy once you express g(x) using f (x).)
3. Fourier series. Find the Fourier series for the function which is 2π-periodic, with f (x) = x
for −π < x < π. By considering the series at x = π/2, deduce the value of the alternating
sum
∞
X
1 1 1
(−1)k
= 1 − + − + ···
2k + 1
3 5 7
k=0
4. Convergence of Fourier series and Fourier sine/cosine series.
(a) By finding an appropriate Fourier series and using the convergence theorem for Fourier
series (see Notes Sec. 11.1), show that
∞
n
8X
sin(2nx), 0 < x < π. · · · (∗).
cos x =
π n=1 (2n)2 − 1
(
− cos x,
(Hint: Extend cos x on 0 < x < π to a function g(x) =
cos x,
−π ≤ x < 0
,
0≤x<π
then find the Fourier series of g(x). )
(b) The series in (a) (the series in the righthand side of (∗)) converges for all x. What function
does it converge to?
(Hint: Read carefully what the theorem for convergence of Fourier series (see Notes Sec.
11.1) means.)
X
n
π
(−1)(n−1)/2 = √ .
(c) Show that
2
4n − 1
8 2
n=1,3,5,...