Lecture 8 Sujin Khomrutai – 1 / 23
Method of Applied Math
Lecture 8: Fourier Series
Sujin Khomrutai, Ph.D.
Introduction
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 2 / 23
In this chapter we study
1. periodic functions, i.e. satisfying f(x + P) = f(x) 2. Fourier series representation
3. Solutions to IVP with a periodic external force.
4. functions on an interval f : [a, b] → R 5. Fourier sine/cosine series
6. Solutions to boundary value problems (BVPs).
Example
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 3 / 23
EX. Forced Vibration
Mathematical model:
my′′ + γy′ + ky = r(t)
Example
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 4 / 23
r(t) is periodic
For such r(t), we will express r(t) as a Fourier series:
r(t) = a0
2 +
∞
X
n=1
(an cos nt + bn sin nt).
Periodic Functions
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 5 / 23
Definition For a function f(x), if there is p > 0 such that f(x + p) = f(x) for all x,
f(x) is called a periodic function and p is a period of f(x), or simply f is p-periodic.
Example 1
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 6 / 23
EX. The function f defined by
f(x) = x for −1 < x < 1, f(x + 2) = f(x) for all x, is 2-periodic.
Example 2
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 7 / 23
EX. Use the well-known identity
cos(A + 2π) = cos A, sin(A + 2π) = sinA
so show that f(x) = cos x and g(x) = sin x are 2π-periodic.
Example 3
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 8 / 23
EX. Generally, for n = 1,2, . . ., show that the functions fn(x) = cos nx and gn(x) = sin nx
are 2π-periodic.
Example 4
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 9 / 23
EX. Let L > 0. Show that f(x) = cos πx
L and g(x) = sin πx L
are 2L-periodic. Prove that the same conclusion holds for
fn(x) = cos nπx
L and gn(x) = sin nπx L , for n = 1, 2, . . .
Question 3
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 10 / 23
Q. Show that
f(x) = sin2 3x
is a 2π-periodic function.
Periodic Functions
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 11 / 23
Note. If f, g are P-periodic functions, then so is af + bg, for any constants a, b. In fact,
(af + bg)(x + P) = af(x + P) + bg(x + P) = af(x) + bg(x)
= (af + bg)(x).
This fact is applied to
a1f1 + a2f2 + · · · Plan
1. P = 2π
2. P = 2L, L > 0 arbitrary.
Fourier Series: Period 2π
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 12 / 23
Definition Given numbers a0, an, bn (n = 1, 2, . . .), a series of the form
a0
2 + (a1 cos x + b1 sin x) + (a2 cos 2x + b2 sin 2x) + · · ·
= a0
2 +
∞
X
n=1
(an cos nx + bn sin nx),
is called a Fourier series with period 2π (the right hand side is 2π-periodic).
a0, an, bn’s are called coefficients.
Fourier Series: Period 2π
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 13 / 23
Theorem Let f(x) be a 2π-periodic function. Assume that 1. f is piecewise continuous, and
2. f′(x−), f′(x+) exist at every x.
Then f can be expanded into a Fourier series:
f(x) = a0
2 +
∞
X
n=1
(an cos nx + bn sin nx).
(See the computation of a0, an, bn below).
This series is called the Fourier series expansion of f(x).
Note. Piecewise continuous means there are finitely many jump discontinuities in one period.
Computation
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 14 / 23
Theorem If f is 2π-periodic with the Fourier series expansion f(x) = a0
2 +
∞
X
n=1
(an cos nx + bn sin nx), then the coefficients are computed from
a0 = 1 π
Z π
−π
f(x)dx an = 1
π
Z π
−π
f(x) cos nx dx (n = 1,2, . . .) bn = 1
π
Z π
−π
f(x) sin nx dx (n = 1,2, . . .)
Example 5
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 15 / 23
EX. Find the Fourier series expansion of
f(x) = (
−k −π < x < 0
k 0 < x < π , f(x + 2π) = f(x).
Example 5
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 16 / 23
Example 6
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 17 / 23
EX. Find the Fourier series expansion of
f(x) = 1 + cos 2x.
ANS.
a0 = 2, a1 = b1 = 0, a2 = 1, b2 = 0, a3 = b3 = · · · = 0
Example 7
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 18 / 23
EX. Find the Fourier series expansion of
f(x) = −2 + sin x − 5 cos 2x.
ANS.
a0 = −4, a1 = 0, b1 = 1, a2 = −5, b2 = 0, a3 = b3 = a4 = b4 = · · · = 0.
Question
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 19 / 23
Q. Use the identity
cos2 A = 1 + cos 2A
2 ,
to find the Fourier series expansion of
f(x) = cos2 x.
Example 8
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 20 / 23
EX. Use the integral formulas Z
x cos nxdx = x sin nx
n + cos nx
n2 + C, Z
x sin nxdx = −x cos nx
n + sin nx
n2 + C,
to find the Fourier series expansion of a 2π-periodic function f where
f(x) = x for −π < x < π.
Convergence of Fourier Series
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 21 / 23
Theorem Let f(x) be a 2π-periodic function. Assume that 1. f is piecewise continuous, and
2. f′(x−), f′(x+) exist at every x.
Then sum of the Fourier series of f satisfies 1. if f(x) is continuous at x0, then
a0
2 +
∞
X
n=1
(an cos(nx0) + bn sin(nx0)) = f(x0);
2. if f is discontinuous at x0, then a0
2 +
∞
X
n=1
(an cos(nx0) + bn sin(nx0)) = f(x−0 ) + f(x+0 )
2 .
Example 9
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 22 / 23
EX. The following figure shows a 2π-periodic function f in one period. Discuss the sum of the Fourier series expansion of f at x0
where
1. x0 ∈ (−π, π)
2. x0 = −π and x0 = π.
Example 10
Introduction Forced vibration Def. Periodic EX 1
EX 2 EX 3 EX 4 Q 3
Fourier Series: 2π Formula
EX 5 EX 6 EX 7 Q 4 EX 8
Convergence EX 9
EX 10
Lecture 8 Sujin Khomrutai – 23 / 23
EX. Given that a 2π-periodic function f where f(x) =
(
−π/4 if −π < x < 0 π/4 if 0 ≤ x < π has the Fourier series expansion
sin x + 1
3 sin 3x + 1
5 sin 5x + · · · , evaluate the sum
1 − 1
3 + 1
5 − 1
7 + · · · .
