Homework 12, Advanced Calculus 1

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Homework 12, Advanced Calculus 1

1. Determine the Fourier series of the following functions on [−π, π], in the formP

n∈Zc_ne^inx: (a) f (x) = 2 + 7 cos(3x) − 4 sin(2x.)

(b) f (x) = x.

(c) f (x) = (π − x)(π + x).

2. Continue from Problem 1, (a) e^x.

(b) e^|x|.

3. Let f be an even function of period 2π with f (x) = cos(2x) for x ∈ [0,^π₂] and f (x) = −1 for x ∈ (^π₂, π).

(a) Complete the definition of f on its domain [−π, π].

(b) Find its Fourier series.

(c) Compute the sum

∞

X

k=1

(−1)^k

(2k + 1)(2k − 1)(2k + 3).

4. Find the Fourier series, in the form a₀+P a_ncos(nx) + b_nsin(nx), of the function

f (x) = (π − |x|)² on [−π, π].

Use your result to prove the identity

∞

X

n=1

1 n² = π²

6 ,

∞

X

n=1

1 n⁴ = π⁴

90.

(Hint: Check that f (x) is an even function. What can you say about f (x) cos(nx) and f (x) sin(nx) and their integrals on [−π, π]?)

5. Given f ∼P

ncne^inx, f differentiable and 2π−periodic, prove that f⁰ ∼P

nincne^inx. 6. Continue from Problem 5, suppose that both f and f⁰ are equal to their Fourier series.

Solve the differential equation

f⁰(x) + 2f (x − π) = sin x.

7. Continue from Problem 5 and 6, suppose in addition that f ∈ C². Find all possible values of a ∈ R so that

f⁰⁰(x) + af (x) = f (x + π), for all x ∈ R and f 6= 0.

8. Rudin Chapter 8 Exercise 13 9. Rudin Chapter 8 Exercise 15 10. Rudin Chapter 8 Exercise 16 11. Rudin Chapter 8 Exercise 19 12. Rudin Chapter 8 Exercise 27

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