HOMEWORK 7 (DEFINITION OF FOURIER SERIES)

1. (a) Show that the functions 1, cosx, cos 2x, · · · sinx, sin 2x, · · · constitute an orthogonal set on the in-terval (−π, π).

(b) Normalize the set.

(Answer:{

1√2π, cosmx√

π, sinnx√

π

}, m,n = 1, 2, · · · )

2. (a) Show that the functions f1(x) = 1 and f2(x) = x are orthogonal on the interval (−1, 1).(b) Determine the constants A and B so that the function f3(x) = 1 +Ax+Bx2 is orthogonal to both

f1 and f2 on (−1, 1).(Answer: A = 0, B = −3)

3. Two continuous functions f(x) and g(x) are linearly independent on an interval (a, b); that is, one isnot a constant times the other. Determine the linear combination f + Ag of those functions which isorthogonal to g.

(Answer: f(x)− (f,g)(g,g)

g(x))

4. Find the Fourier series on the interval (−π, π) for the following functions.(a) f(x) = x when −π < x < π. Also, note the sum of the series when x = ±π.

(Answer: 2∑∞

n=1(−1)n+1

nsinnx)

(b) f(x) =

{1 when − π < x < 0

2 when 0 < x < π

(Answer: 32

+ 2π

∑∞n=1

sin(2n−1)x2n−1 )

5. (a) Find the Fourier series on the interval (−π, π) for f(x) =

{0 when − π ≤ x ≤ 0

sinx when 0 ≤ x ≤ π

(Answer: 1π

+ 12

sinx− 2π

∑∞n=1

cos 2nx4n2−1 )

(b) Prove that the Fourier series found in part(a) converges uniformly for all −∞ < x <∞.(c) Sketch the graph of the function represented by the series for all −∞ < x <∞.

6. Find (i) the Fourier cosine series and (ii) the Fourier sine series, corresponding to the following functionsdefined on the interval (0, π). Also sketch the graph of the functions represented by these series for all−∞ < x <∞.(a) f(x) = 1, 0 < x < π

(Answer: (i) 1 (ii) 4π

∑∞n=1

sin(2n−1)x2n−1 )

(b) f(x) = π − x, 0 < x < π

(Answer: (i) π2

+ 4π

∑∞n=1

cos(2n−1)x(2n−1)2 (ii) 2

∑∞n=1

sinnxn

)

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