010.141
FOURIER SERIES, INTEGRALS, AND TRANSFORMS
MODULE 3
ADVANCED ENGINEERING MATHEMATICS
2B.D. Youn2010 Engineering Mathematics II Module 2
Fourier Series, Integrals, and Transform
Ø Fourier series: Infinite series designed to represent general periodic functions in terms of simple ones (e.g., sines and cosines).
Ø Fourier series is more general than Taylor series because many discontinuous periodic functions of practical interest can be developed in Fourier series.
Ø Fourier integrals and Fourier Transforms extend the ideas and techniques of Fourier series to non-periodic functions and have basic applications to PDEs.
3B.D. Youn2010 Engineering Mathematics II Module 2
PERIODIC FUNCTIONS
Ø A periodic function is a function such that f(x + p) = f(x) where p is a period.
Ø The smallest period is called a fundamental period.
4B.D. Youn2010 Engineering Mathematics II Module 2
FOURIER SERIES
Assume that f(x) is periodic with period 2p and is integrable over a period.
f(x) can be represented by a trigonometric series:
i.e., the series converges and has sum f(x)
( ) ( )f x = a + a cos nx + b sin nx0 n nn = 1
¥
å
5B.D. Youn2010 Engineering Mathematics II Module 2
Euler Formulas
Fourier coefficients
a0, an, bn are the Fourier coefficients.
The corresponding series is the Fourier series
( )
( )
( )
01
2
1 cos
1 sin
n
n
a = f x dx
a = f x nx dx
b = f x nx dx
p
p
p
p
p
p
p
p
p
-
-
-
×
×
ò
ò
ò
TRIGONOMETRIC SERIES WITH A PERIOD OF 2p
6B.D. Youn2010 Engineering Mathematics II Module 2
( )f x = k, < x < 0
k, < x < - -ì
íî
pp0
Example: Rectangular wave
FOURIER SERIES (cont)
( ) [ ]
0
00
n
1 1a = k dx+ dx 02 2
1 1a = cos x dx= 0 =0
k
f x n
p
p
p
p
p p
p p
-
-
- =ò ò
ò
Fourier Coefficients( )
( )
( )
n
0
0
0
0
1b = sin x dx
1 1= sin x dx + sin x dx
s nx cos nx= n n
4k= 1 1 1 1 = odd n
f x n
k n k n
k co
k for nn
p
p
p
p
p
p
p
p p
p
p p
-
-
-
-
æ öé ù é ù+ -ç ÷ê ú ê úç ÷ë û ë ûè ø
+ + +
ò
ò ò
7B.D. Youn2010 Engineering Mathematics II Module 2
EXAMPLE 1: RECTANGULAR WAVE
( )
( )
( ) ( )
0 n nn = 1 n = 1
2n+1n = 0
n = 0
x =a + a cos cos
= sin 2 1
4 sin 2 12 1
4 1 1sinx+ sin 3 sin 52 1+1 5
f nx b nx
b n x
k n xn
k x x
p
p
¥ ¥
¥
¥
+
+
= ++
æ ö= + +ç ÷×è ø
å å
å
å
L
Fourier series:
8B.D. Youn2010 Engineering Mathematics II Module 2
ORTHOGONALITY OF SINE, COSINE FUNCTIONS
Two functions fm and fn of the same form are orthogonal if ò fm fn dx = 0 for all m ¹ n and ò fm fn dx = a for all m = n.
( )
( )
2
2
cos mx cos nx dx 0 for all m n
cos nx dx 0 for all m n
sin mx sin nx dx 0 for all m n
sin nx dx 0 for all m n
cos mx sin nx dx
p
p
p
p
p
p
p
p
p
p
-
-
-
-
-
= ¹ ±
¹ =
= ¹
¹ =
ò
ò
ò
ò
ò 0 for all m n= =
9B.D. Youn2010 Engineering Mathematics II Module 2
CONVERGENCE AND SUM OF FOURIER SERIES
10B.D. Youn2010 Engineering Mathematics II Module 2
CONVERGENCE AND SUM OF FOURIER SERIES
Proof: For continuous f(x) with continuous first and second orderderivatives.
( )
( )
( )
n
0
1a cos nx dx
f sin nx1 1 sin nxdx dx
f sin nx1 1 'sin nx dx
1 'sin nx dx
f x
xf
n n
xf
n n
fn
p
p
p p
p p
p p
pp
p
p
p
p p
p p
p
-
- -
--
-
=
¢æ ö
¢= -ç ÷è ø
= -
= -
ò
ò ò
ò
ò
1442443
11B.D. Youn2010 Engineering Mathematics II Module 2
Repeating the process:
Since f" is continuous on [– π, π ]
CONVERGENCE AND SUM OF FOURIER SERIES (cont)
( ) dx nxcosx"fn1a 2n ò
p
p-p-=
( )
( )
22n
22n
nM2
nM2a
dxMn1dx nxcosx"f
n1a
Mx"f
=pp
<
p<
p=
<
òòp
p-
p
p-
12B.D. Youn2010 Engineering Mathematics II Module 2
Similarly for
Hence
which converges. (see Page 670)
CONVERGENCE AND SUM OF FOURIER SERIES (cont)
2n nM2b <
( ) ÷øö
çèæ +++++++< L22220 3
131
21
2111M2axf
13B.D. Youn2010 Engineering Mathematics II Module 2
FUNCTIONS OF ANY PERIODP = 2L
14B.D. Youn2010 Engineering Mathematics II Module 2
FUNCTIONS OF ANY PERIOD (cont)P = 2L
( )
( ) ( ) ( )
( ) ( ) ( )òòò
òòò
åå
--
p
p-
--
p
p-
¥
=
¥
=
=p
p=
p=
p=
p×
pp
=p
=
++=
p=
L
L
L
L0
L
L
L
Ln
1 n n
1 n n0
dxxf L21 dx
Lxf
21 dvvg
21 a
dxL
xncosxf L1 dx
L
Lxncosxf 1 dv nvcosvg 1 a
nvsinb nvcosa a vg
Lxv
Proof: Result obtained easily through change of scale.
Same for a0, bn
15B.D. Youn2010 Engineering Mathematics II Module 2
ANY INTERVAL (a, a + P)P = PERIOD = 2L
( )
( )
( )
( ) dx P
x2nsinxf P2 b
dx P
x2ncosxf P2 a
dxxf P1 a
Px2ninsb
Px2ncosa a xf
P a
an
P a
an
P a
a0
1 n nn0
p=
p=
=
÷øö
çèæ p+
p+=
ò
ò
ò
å
+
+
+
¥
=
a a + P
16B.D. Youn2010 Engineering Mathematics II Module 2
EXAMPLE OF APPLICATION
Find the Fourier series of the function
Solution:
( )ïî
ïí
ì
17B.D. Youn2010 Engineering Mathematics II Module 2
EXAMPLE OF APPLICATION (cont)
3ncos
nncos12 b
3nins
nncos12 a
0 a
3ncos
37nsco
3ncos ncos
34ncos
3nins
37nsin
3nins ncos
34nsin
n
n
0
p÷øö
çèæ
pp-
=
p÷øö
çèæ
pp-
-=
=
p=
ppp=
p
p=
ppp=
p
x-3 71 4
f(x)
18B.D. Youn2010 Engineering Mathematics II Module 2
EVEN AND ODD FUNCTIONS
Examples: x4, 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.
19B.D. Youn2010 Engineering Mathematics II Module 2
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
EVEN AND ODD FUNCTIONS
( ) ( )
( ) 0dx xh
dx xg2dx xg
L
L
L
0
L
L
=
=
ò
òò
-
-
20B.D. Youn2010 Engineering Mathematics II Module 2
EVEN AND ODD FUNCTIONS
21B.D. Youn2010 Engineering Mathematics II Module 2
EVEN AND ODD FUNCTIONS
22B.D. Youn2010 Engineering Mathematics II Module 2
SUM
23B.D. Youn2010 Engineering Mathematics II Module 2
SUM
24B.D. Youn2010 Engineering Mathematics II Module 2
SUM
25B.D. Youn2010 Engineering Mathematics II Module 2
SUM
26B.D. Youn2010 Engineering Mathematics II Module 2
Forced Oscillations
27B.D. Youn2010 Engineering Mathematics II Module 2
Forced Oscillations
28B.D. Youn2010 Engineering Mathematics II Module 2
Forced Oscillations
29B.D. Youn2010 Engineering Mathematics II Module 2
Forced Oscillations
30B.D. Youn2010 Engineering Mathematics II Module 2
Forced Oscillations
31B.D. Youn2010 Engineering Mathematics II Module 2
Consider a function f(x), periodic of period 2π. Consider an approximation of f(x),
The total square error of F
is minimum when F's coefficients are the Fourier coefficients.
APPROXIMATION BYTRIGONOMETRIC POLYNOMIALS
( ) 0n 1
( ) x cos sinN
n nf x F A A nx B nx=
» = + +å
( )òp
p-
-= dxFf E 2
Read Page 503 to see the derivation procedure of Eq. (6)
32B.D. Youn2010 Engineering Mathematics II Module 2
PARSEVAL'S THEOREM
( ) ( )2 2 2 20n 1
12 n na a b f x dxp
pp
¥
= -
+ + £å ò
The square error, call it E*, is
Where an, bn are the Fourier coefficients of f.
33B.D. Youn2010 Engineering Mathematics II Module 2
FOURIER INTEGRALS
Since many problems involve functions that are nonperiodic and are ofinterest on the whole x-axis, we ask what can be done to extend the methodof Fourier series to such functions. This idea will lead to “Fourier integrals.”
34B.D. Youn2010 Engineering Mathematics II Module 2
FOURIER INTEGRALS
35B.D. Youn2010 Engineering Mathematics II Module 2
FOURIER INTEGRALS
36B.D. Youn2010 Engineering Mathematics II Module 2
FOURIER COSINE AND SINE INTEGRALS
37B.D. Youn2010 Engineering Mathematics II Module 2
FOURIER COSINE AND SINE INTEGRALS
38B.D. Youn2010 Engineering Mathematics II Module 2
FOURIER COSINE AND SINE INTEGRALS
( ) ( ) ( )0
f x cos sin dA x B xw w w w w¥é ù= +ë ûò
( )
( )
( )
01
2
1 cos
1 sin
n
n
a = f x dx
a = f x nx dx
b = f x nx dx
p
p
p
p
p
p
p
p
p
-
-
-
×
×
ò
ò
ò
( ) ( )f x = a + a cos nx + b sin nx0 n nn = 1
¥
å
( ) ( ) ( )
( ) ( ) ( )
1 cos d
1 sin d
A f v v v
B f v v v
w wp
w wp
¥
-¥
¥
-¥
=
=
ò
ò
39B.D. Youn2010 Engineering Mathematics II Module 2
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.
Where f(x) is discontinuous, the F.I. equals the average of the left-hand and right-hand limit of f(x).
( ) ( )òò ¥®¥-® +b
0 b
0
a a
dxxf limdxxf lim
40B.D. Youn2010 Engineering Mathematics II Module 2
For an even or odd function the F.I. becomes much simpler.
If f(x) is even
If f(x) is odd
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( ) integral sineFourier dω ωx sinωB xf
dv ωvsin vfπ2 ωB
integral cosineFourier dω ωx cosωA xf
dv ωv cos vfπ2 ωA
0
0
ò
ò
ò
ò
¥
¥
¥-
¥
¥
¥-
=
=
=
=
FOURIER COSINE AND SINE INTEGRALS
41B.D. Youn2010 Engineering Mathematics II Module 2
EXAMPLE
Consider
Evaluate the Fourier cosine integral A(w) and sine integral B(w).
( ) 0 k 0, x e xf kx >>= -
Integration by parts gives
For Fourier cosine integral,
Fourier cosine integral is
42B.D. Youn2010 Engineering Mathematics II Module 2
EXAMPLE
Integration by parts gives
For Fourier sine integral,
Fourier sine integral is
Laplace integrals
43B.D. Youn2010 Engineering Mathematics II Module 2
FOURIER SINE AND COSINE TRANSFORMS
For an even function, the Fourier integral is the Fourier cosine integral
( ) ( ) ( ) ( ) ( ) ( ) ˆπ2 dv ωv cos vf
π2 ωAdω ωx cosωA xf
0
wcf=== òò¥
¥-
¥
( ) ( ) ( ) ( )
( ) ( ) ( )
( ).ˆ of transformcosineFourier inverse theis
. of transformcosineFourier theas defined is ˆ
dωx cos ˆπ2 dωx cos ˆ
π2 xf
dv ωv cos vfπ2 dv ωv cos vf
π2
2π ωA
2π ˆ
00
00
c
c
cc
c
ff
ff
ff
f
w
wwww
w
òò
òò¥¥
¥¥
==
===
Then
44B.D. Youn2010 Engineering Mathematics II Module 2
FOURIER SINE AND COSINE TRANSFORMS (cont)
Similarly for odd function
( ) ( )
( ) ( ) dωx sin f̂π2 xf T sine F Inverse
dxωx sin xfπ2 f̂ T sine F
0c
0c
ò
ò
¥
¥
ww=®
=w®
45B.D. Youn2010 Engineering Mathematics II Module 2
NOTATION AND PROPERTIES
{ } { }{ } { }
{ } { } { }{ } { } { }
( ){ } { } ( )
( ){ } { }
{ } { } ( )
{ } { } ( )0fω π2fω f(6)
0f π2fω f(5)
fω xf(4)
0f π2fω xf(3)
gb fa bgaf(2)gb fa bgaf(1)
ˆ , ˆ
ˆ ,ˆ
s2
s
c2
c
cs
sc
sss
ccc
s1
sc1
c
sscc
¢+-=¢¢
¢--=¢¢
-=¢
-=¢
+=++=+
==
==--
FF
FF
FF
FF
FF
FF
FFFFFF
ffff
ffff
46B.D. Youn2010 Engineering Mathematics II Module 2
FOURIER TRANSFORM
( ) ( ) ( )[ ]
( ) ( )4444 34444 21
w
¥
¥-
¥
¥
¥-
¥
òò
òò
w-wwp
=
ww+wwwp
=
in Even
0
0
vdx vcos vfd1
vdxsinvsin vfxcosvcos vfd1 xf
( ) ( ) ( )[ ]
( ) ( )
( ) ( )ò
ò
ò
¥
¥-
¥
¥-
¥
=w
=w
+=
vd ωvsin vfπ1 B
vd ωv cos vfπ1 A
dx ωxsin ωBωx cosωAxf0
The F.I. is:
Replacing
47B.D. Youn2010 Engineering Mathematics II Module 2
FOURIER TRANSFORM
( ) ( ) ( )
( ) ( ) ( ) lvd v xsinvfd21 xf
vd v xcosvfd21 xf
in odd4444 34444 21
w
¥
¥-
¥
¥-
¥
¥-
¥
¥-
òò
òò
-wwp
=
-wwp
=
Now,
( ) ( ) ( )
( ) ( ) IntegralFourier Complex dve vfd21
dω GiωF21 xf
vxiωòò
ò
¥
¥-
-¥
¥-
¥
¥-
wp
=
w+p
=
48B.D. Youn2010 Engineering Mathematics II Module 2
FOURIER TRANSFORM (cont)
( ) ( )
( )
( ) ( ) wwp
=
pw
p=
ò
òò
¥
¥-
ºw
¥
¥-
-¥
¥-
de f̂21 xf
dve vf21de
21 xf
xiω
f of TransformFourier f̂
viωxiω
444 3444 21
49B.D. Youn2010 Engineering Mathematics II Module 2
NOTATION AND PROPERTIES
{ } { } { }{ } { }
{ } { }{ } { } { }g f2 gf
fω f
fiω f
gb fa bgaf
2
FFF
FFF
+p=*
-=¢¢
=¢
+=+
FF
FF