ENGINEERING MATHEMATICS II
010.141
Byeng Dong Youn (
윤병동윤병동윤병동윤병동)
CHAP. 11
Fourier Series, Integrals, and
Transforms
ABSTRACT OF CHAP. 11
Fourier Analysis in Chap. 11 concerns periodic
phenomena—thinking of rotating parts of machines, alternating electric currents, or the motion of planets.
However, the underlying ideas can also be extended to non- periodic phenomena.
Fourier series: Infinite series designed to represent general periodic functions in terms of simple ones (e.g., sines and cosines).
Fourier series is more general than Taylor series because many discontinuous periodic functions of practical interest can be
developed in Fourier series.
Fourier integrals and Fourier Transforms extend the ideas and
techniques of Fourier series to non-periodic functions and have basic
CHAP. 11.1
FOURIER SERIES
Infinite series designed to represent general periodic
functions in terms of simple ones like cosines and sines.
PERIODIC FUNCTIONS
PERIODIC FUNCTIONS
FOURIER SERIES
EXAMPLE
EXAMPLE
CONVERGENCE AND SUM OF FOURIER SERIES
Proof: For continuous f(x) with continuous first and second order derivatives.
( ) ( ) ( )
n
0
a 1 cos nx dx
f sin nx
1 1 sin nx
dx dx
f sin nx
1 1
'sin nx dx
1 'sin nx dx f x
x f
n n
x f
n n
f
π
π
π π
π π
π π
π π
π
π
π π
π π
−
− −
− −
=
′
= − ′
= −
= −
∫
∫ ∫
∫
∫
1442443
Repeating the process:
Since f" is continuous on [– π , π ]
CONVERGENCE AND SUM OF FOURIER SERIES (cont)
( ) x cos nx dx
"
f n
a 1
n 2 ∫
π
π
π −
−
=
( ) ( )
2 n 2
2 n 2
n M 2 n
M a 2
dx M n
dx 1 nx cos x
"
f n
a 1
M x
"
f
= π
< π
π
<
π
=
<
∫
∫
π
π
− π
π
−
Similarly for
Hence
which converges.
CONVERGENCE AND SUM OF FOURIER SERIES (cont)
n 2
n M b < 2
( )
+ + + + + +
+
< L
2 2
2 0 2
3 1 3
1 2
1 2
1 1 1 M 2 a
x
f
HOMEWORK IN 11.1
HW1. Problem 9
HW2. Problem 15
CHAP. 11.2
FUNCTIONS OF ANY PERIOD P=2L
general periodic functions with P=2L.
FUNCTIONS OF ANY PERIOD
P = 2L
EXAMPLE
FUNCTIONS OF ANY PERIOD (cont) P = 2L
( )
( ) ( ) ( )
( ) ∫ ( ) ∫ ( )
∫
∫
∫
∫
∑
∑
−
− π
π
−
−
− π
π
−
∞
=
∞
=
π =
= π
= π
= π
⋅ π π
= π
= π
+ +
=
= π
L
L L
L 0
L
L L
L n
1 n
n 1
n
n 0
dx x L f
2 dx 1 x L
2 f dv 1 v 2 g
a 1
L dx x cos n
x L f
dx 1 L L
x cos n
x 1 f
dv nv cos v 1 g
a
nv sin b nv
cos a
a v g
L v x
Proof: Result obtained easily through change of scale.
Same for a
0, b
nANY INTERVAL (a, a + P) P = PERIOD = 2L
( )
( ) ( )
( ) dx
P x sin 2n
x P f
b 2
P dx x cos 2n
x P f
a 2
dx x P f
a 1
P x in 2n
s P b
x cos 2n
a a
x f
P a
a n
P a
a n
P a
a 0
1 n
n n
0
= π
= π
=
π
π + +
=
∫
∫
∫
∑
+ + +
∞
=
a a + P
HOMEWORK IN 11.2
HW1. Problem 3
HW2. Problem 5
CHAP. 11.3
EVEN AND ODD FUNCTIONS. HALF- RANGE EXPANSIONS
Take advantage of function properties.
Examples: x
4, cos x
f(x) is an even function of x, if f(-x) = f(x). For example, f(x) = x sin(x), then
f(- x) = - x sin (- x) = f(x)
and so we can conclude that x sin (x) is an even function.
EVEN AND ODD FUNCTIONS
Properties
1. If g(x) is an even function
2. If h(x) is an odd function
3. The product of an even and odd function is odd
( ) ( )
( ) x dx 0 h
dx x g 2 dx x g
L
L
L
0 L
L
=
=
∫
∫
∫
−
−
EVEN AND ODD FUNCTIONS
THEOREMS
EXAMPLE
HOMEWORK IN 11.3
HW1. Problem 11
HW2. Problem 15
CHAP. 11.5
FORCED OSCILLATIONS
Connections with ODEs and PDEs.
Forced Oscillations
Forced Oscillations
Forced Oscillations
Forced Oscillations
Forced Oscillations
HOMEWORK IN 11.5
HW1. Problem 6
CHAP. 11.6
APPROXIMATION BY
TRIGONOMETRIC POLYNOMIALS
Useful in approximation theory.
Consider a function f(x), periodic of period 2 π . Consider an approximation of f(x),
The total square error of F
is minimum when F's coefficients are the Fourier coefficients.
APPROXIMATION BY
TRIGONOMETRIC POLYNOMIALS
( )
0n 1
( ) x cos sin
N
n n
f x F A A nx B nx
=
≈ = + ∑ +
( )
∫
π
π
−
−
= f F dx
E 2
PARSEVAL'S THEOREM
( ) ( )
2 2 2 2
0
n 1
2 a a
nb
n1 f x dx
π
π π
∞
= −
+ ∑ + ≤ ∫
The square error, call it E*, is
where a
n= A
nand b
n= B
n.
Parseval’s theorem:
HOMEWORK IN 11.6
HW1. Problem 2
CHAP. 11.7
FOURIER INTEGRAL
Extension of the Fourier series method to non-periodic
functions.
FOURIER INTEGRALS
Since many problems involve functions that are nonperiodic and are of
interest on the whole x-axis, we ask what can be done to extend the method
of Fourier series to such functions. This idea will lead to “Fourier integrals.”
FOURIER INTEGRALS
FOURIER INTEGRALS
FOURIER COSINE AND SINE INTEGRALS
FOURIER COSINE AND SINE INTEGRALS
FOURIER COSINE AND SINE INTEGRALS
( ) ( ) ( )
f x = ∫
0∞ A ω cos ω x + B ω sin ω x d ω
( ) ( )
( )
0
1 2
1 cos
1 sin
n
n
a = f x dx
a = f x nx dx
b = f x nx dx
π
π π
π π
π
π π π
−
−
−
⋅
⋅
∫
∫
∫
( ) ( )
f x = a + 0 a cos nx + b sin nx n n
n = 1
∑ ∞
( ) ( ) ( )
( ) ( ) ( )
1 cos d
1 sin d
A f v v v
B f v v v
ω ω
π
ω ω
π
∞
−∞
∞
−∞
=
=
∫
∫
THEOREM1: EXISTENCE
If f(x) is piecewise continuous in every finite interval and has a right hand and left hand derivative at every point and if
exists, then f(x) can be represented by the Fourier integral.
The F.I. equals the average of the left-hand and right-hand limit of f(x) where f(x) is discontinuous.
( ) ∫ ( )
∫ → ∞
∞
−
→
+
b b 0
0
a lim a f x dx lim f x dx
For an even or odd function the F.I. becomes much simpler.
If f(x) is even
If f(x) is odd
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
0
0
0
0
A ω 2 f v cos ω v dv π
f x A ω cos ω x d ω Fourier cosine integral
B ω 2 f v sin ω v dv π
f x B ω sin ω x d ω Fourier sine integral
∞
∞
∞
∞
=
=
=
=
∫
∫
∫
∫
FOURIER COSINE AND SINE INTEGRALS
EXAMPLE
Consider
Evaluate the Fourier cosine integral A(ω) and sine integral B(ω).
( ) x e x 0, k 0
f = − kx > >
Integration by parts gives For Fourier cosine integral,
Fourier cosine integral is
EXAMPLE
Integration by parts gives For Fourier sine integral,
Fourier sine integral is
Laplace integrals
HOMEWORK IN 11.7
HW1. Problem 1
HW2. Problem 7
CHAP. 11.8
FOURIER COSINE AND SINE TRANSFORMS
An integral transform is a transformation in the form of an integral that produces from given functions new
functions depending on a different variable.
FOURIER SINE AND COSINE TRANSFORMS
For an even function, the Fourier integral is the Fourier cosine integral
( ) ( ) ( ) ( ) ( ) ( )
0
2 2 ˆ
f x A ω cos ω x d ω A ω f v cos ω v dv
π π f
cω
∞ ∞
− ∞
= ∫ = ∫ =
( ) ( ) ( ) ( )
( ) ( ) ( )
( )
0 0
0 0
π π 2 2
ˆ A ω f v cos ω v dv f v cos ω v dv
2 2 π π
2 ˆ 2 ˆ
f x cos ω x d cos ω x d
π π
ˆ is defined as the Fourier cosine transform of . is the inverse Fourier cosine transform o
c
c c
c
f
f f
f f
f ω
ω ω ω ω
ω
∞ ∞
∞ ∞
= = =
= =
∫ ∫
∫ ∫
f . f ˆ
cThen
{ }
FOURIER SINE AND COSINE TRANSFORMS (cont)
Similarly for odd function
( ) ( )
( ) ( )
0
0
ˆ 2
sine f x sin ω x dx
π
2 ˆ
sine f x sin ω x d
π
s
s
F T f
Inverse F T f
ω
ω ω
∞
∞
→ =
→ =
∫
∫
{ }
1{ }
s
f , = f ˆ
s s−f ˆ
s= f
F F
NOTATION AND PROPERTIES
{ } { }
{ } { }
{ } { } { }
{ } { } { }
( )
{ } { } ( )
( )
{ } { }
{ } { } ( )
{ } { } 2 ( )
0 π f
f 2 f ω
(5)
f x ω
f (4)
0 π f
f 2 x ω
f (3)
g b
f a
bg af
(2)
g b
f a
bg af
(1)
ˆ ˆ ,
ˆ ˆ ,
c 2 c
c s
s c
s s
s
c c
c
s 1 s c
1 c
s s
c c
− ′
−
′′ =
−
′ =
−
′ =
+
= +
+
= +
=
=
=
=
−
−
F F
F F
F F
F F
F F
F F
F
F F
F
f f
f f
f f
f
f
TRANSFORMS OF DERIVATIVES
FOURIER TRANSFORM
( ) ( ) [ ]
( ) ( )
0
1 cos cos sin sin
1 cos
2
f x f v v x v x dv d
f v x v dv d
∞ ∞
− ∞
∞ ∞
= +
= −
∫ ∫
∫ ∫
ω ω ω ω ω
π
ω ω ω
π
( ) ( ) ( )
( ) ( )
( ) ( )
0
cos sin
1 cos
1 sin
f x A x B x dx
A f v v dv
B f v v dv
∞
∞
− ∞
∞
− ∞
= +
=
=
∫
∫
∫
ω ω ω ω
ω ω
π
ω ω
π
The F.I. is:
Replacing
FOURIER TRANSFORM
( ) ( ) ( )
( ) ( ) ( )
( ) odd function in
1 cos
2
1 sin 0
2
F
f x f v x v dv d
g x f v x v dv d
∞ ∞
−∞
− ∞
∞ ∞
−∞
− ∞
=
= −
= − =
∫ ∫
∫ ∫
14444 4244444 3
ω ω
ω ω
π
ω ω
π
Now,
( ) ( ) ( )
( ) ( )
1 2
1 Complex Fourier Integral
2
i x v
f x F i G d
f v e dv d
∞
−∞
∞ ∞
−
− ∞ −∞
= +
=
∫
∫ ∫
ωω ω ω
π
π ω
FOURIER TRANSFORM (cont)
( ) ( )
( )
( ) ( )
ˆ Fourier Transform of
1 1
2 2
1 ˆ
2
i v i x
f f
i x
f x f v e dv e d
f x f e d
∞ ∞
−
−∞ − ∞
≡
∞
− ∞
=
=
∫ ∫
∫
14444244443
ω ω
ω
ω
π π ω
ω ω
π
NOTATION AND PROPERTIES
{ } { } { }
{ } { }
{ } { }
{ f g } 2 { } f { } g f
f ω
f i ω
f
g b
f a
bg af
2
F F
F
F F
F
+ π
=
∗
−
′′ =
′ =
+
= +
F F
F
F
