12.2 Fourier Series Trigonometric Series , sin , sin , sin , cos , cos , cos , 1 3 2 3 2 x x x x x x p p p p p p 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 x b x a a x FS p n n n p n n sin cos 2 ) ( 1 0 p p p n n xdx x f p a cos ) ( 1 p p p n n xdx x f p b sin ) ( 1 p p dx x f p a ) ( 1 0 Fourier series of the function f Fourier coefficients of f
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12.2 Fourier Series Trigonometric Series is orthogonal on the interval [ -p, p]. In applications, we are interested to expand a function f(x) defined on.
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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