01. fourier series
TRANSCRIPT
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Eric Augustus J. Tingatinga, Ph.D.
Institute of Civil Engineering
University of the Philippines, Diliman
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A Fourier*seriesmay be defined as adecompositionof a function or any timeseries into the sum or integral of harmonic
waves(sines and cosines) of differentfrequencies.
*Joseph Fourier(1768-1830) made important contributions to the study of
heat conduction.
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sin 2m!tT
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'T/2
T/2
( sin 2n!tT
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%&dt= T
2"mn
1'"m0( ),
cos2m!t
T
!
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'T/2
T/2
( cos2n!t
T
!
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%&dt=
T
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cos2m!t
T
!
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( sin2n!t
T
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f(t) = a0 + ancos!nt,n=1
!
"
f(t) = bnsin!nt,
n=1
!
"
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`'(62') A
a1+1. &)5
"01+19'+1
,)1(2
coscos2
sin2sin1
,0cos1
1
00
0
+
!
!
!=
"#$
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+!=
==
==
(
((
(
n
n
n
n
ntdtnttn
ntdttntdttb
ntdtta
))
))
)
)
)
)
))
)
T = 2!f t( ) = t
f(t) = t= 2 sin t!sin 2t
2+
sin3t
3!!+ (!1)n+1
sin nt
n+!
"
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`'(62')
Figure shows f(t)for the sum of 4, 6, and 10 terms
of the series.
f(t) = t= 2 sin t!sin 2t
2+
sin3t
3!!+ (!1)n+1
sin nt
n+!
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`'(62') D
a0 =
1
2!hdx
0
!
! =h
2,
an =
1
!
hcosnx dx0
!
! = 0
bn =
1
!
hsinnxdx0
!
! =h
n!(1" cosn!) =
2h
n!, n = odd
0, n = even.
#$%
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f(x) =h
2+
2h
!
(sinx
1+
sin3x
3+
sin5x
5+!).
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`'(62') D
Gibbs Phenomenon. The peculiar manner in which theFourier series of a piecewise continuously differentiable
periodic function behaves at a jump discontinuity.
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