12.2 Fourier Series
Trigonometric Series
,sin,sin,sin
,cos,cos,cos,1
32
32
xxx
xxx
ppp
ppp
is orthogonal on the interval [ -p, p].
In applications, we are interested to expand a function f(x) defined on [-p, p] as a linear combination
xbxaa
xFSp
nn
np
nn
sincos2
)(1
0
p
p p
nn xdxxf
pa cos)(
1
p
p p
nn xdxxf
pb sin)(
1
p
pdxxf
pa )(
10
Fourier series of the function f
Fourier coefficients of f
12.2 Fourier Series
xx
xxf
0
00)(
Example:
xbxaa
xfp
nn
np
nn
sincos2
)(1
0
p
p p
nn xdxxf
pa cos)(
1
p
p p
nn xdxxf
pb sin)(
1
p
pdxxf
pa )(
10
Fourier seriesExpand in a Fourier series
2
)1(1
na
n
n
nbn
120
a
nxnxxFSn
nn
n
sincos4
)( 1
1
)1(12
12.2 Fourier Series
Example:
xbxaa
xfp
nn
np
nn
sincos2
)(1
0
p
p p
nn xdxxf
pa cos)(
1
p
p p
nn xdxxf
pb sin)(
1
p
pdxxf
pa )(
10
Fourier seriesExpand in a Fourier series
nb
n
n
)1(1 0na
10 a
x
xxf
01
00)(
)sin())1(1(
2
1)(
1
nxn
xFSn
n
Convergence of a Fourier Series
f(x) is piecewise continuous on the interval [-p,p]; if f(x) is continuous except at a finite number of points in the interval and have only finite discontinuities at these points.
piecewise continuous
pp
Theorem 12.2.1 Conditions for Convergence
' , ff piecewise continuous on [-p,p]
is a point of continuity.
is a point of discontinuity.
denote the limit of f at x from the right and from the left
12.2 Fourier Series
xx
xxf
0
0)(
Example: Expand in a Fourier series
nxnxxFSn
nn
n
sincos4
)( 1
1
)1(12
)()( xfxFS ),0()0,( x
2
)0()0()0( ffFS 0x
2
Remark:
Sequence of Partial Sums
-3 -2 -1 1 2 3
0.2
0.4
0.6
0.8
1
3
)3sin(2)sin(2
2
1 xx
-3 -2 -1 1 2 3
0.2
0.4
0.6
0.8
1
7
)7sin(2
5
)5sin(2
3
)3sin(2)sin(2
2
1 xxxx
-3 -2 -1 1 2 3
0.2
0.4
0.6
0.8
1
-3 -2 -1 1 2 3
0.2
0.4
0.6
0.8
1
x
xxf
01
00)(
)sin())1(1(
2
1)(
1
nxn
xFSn
n
Example:15
ter
ms
25 t
erm
s
Sequence of Partial Sums
-3 -2 -1 1 2 3
0.2
0.4
0.6
0.8
1
-3 -2 -1 1 2 3
0.2
0.4
0.6
0.8
1
x
xxf
01
00)(
)sin())1(1(
2
1)(
1
nxn
xFSn
n
Example:
15 t
erm
s
25 t
erm
s
-3 -2 -1 1 2 3
0.2
0.4
0.6
0.8
1
-3 -2 -1 1 2 3
0.2
0.4
0.6
0.8
1
125
term
s
1000
ter
ms
MATHEMATICA Plot[0.5+Sum[ (1-(-1)^n)*Sin[n x]/(n Pi),{n,1,1000}],{x,-Pi,Pi}];
Periodic Extension
xx
xxf
0
00)(
Example: Consider the funciion
Periodic extension of the function f
3 55 3
12.2 Fourier Series
Example: Consider the function
x
xxf
01
00)(
3 435
Periodic extension of the function f
12.2 Fourier Series
Example: Consider the function
x
xxf
01
00)(
3 435
Periodic extension of the function
)sin())1(1(
2
1)(
1
nxn
xFSn
n
a Fourier series not only represents the function on the interval ( -p, p) but also gives the periodic extension of f outside this interval.
3 435
)(xFS
2p is the fundamental period
)(xf
Periodic Extension
xx
xxf
0
00)(
Example: Consider the funciion
3 55 3
nxnxxFSn
nn
n
sincos4
)( 1
1
)1(12
3 55 3
3 55 3
(A)
(B)
(C)
Which one represents FS(x) ?