082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

23.5 Fourier series 733
Table23.1
Someusefulintegral identities.
∫ T
sin 2nπt dt = 0 forallintegersn
0 T
∫ T
0
∫ T
0
∫ T
0
∫ T
0
∫ T
0
cos 2nπt
T
cos 2nπt
T
cos 2mπt
T
sin 2mπt
T
sin 2mπt
T
dt =0
dt=T n=0
cos 2nπt
T
sin 2nπt
T
cos 2nπt
T
n=1,2,3,...
{ 0 m≠n
dt =
T/2 m=n≠0
{ 0 m≠n
dt =
T/2 m=n≠0
dt = 0
forall integersmandn
23.5 FOURIERSERIES
We have seen that the functions sinωt, sin2ωt, sin3ωt,...,cos ωt, cos2ωt,... are
periodic.Furthermore,linearcombinationsofthemarealsoperiodic.Theyarealsoconvenient
functions to deal with because they can be easily differentiated, integrated, etc.
Theyalsopossessanotherveryusefulproperty--thatof completeness.Thismeansthat
almost any periodic function can be expressed as a linear combination of them and no
additional functions are required to do this. In other words, they can be used as buildingblockstoconstructperiodicfunctionssimplybyaddingparticularmultiplesofthem
together.
We shall see, for example, that the sawtooth waveform with period 2π, shown in
Figure 23.12, isgiven by the particular combination
f(t)=2
(sint− 1 2 sin2t + 1 3 sin3t − 1 4 sin4t + 1 sin5t−···)
5
Thisisaninfiniteserieswhichcanbeshowntoconvergeforalmostallvaluesoft tothe
function f.Thismeansthatifanyvalueoft issubstitutedintotheinfiniteseriesandthe
series is summed, the result will be the same as the value of the sawtooth function at
thatvalueoft.Thereisanexception:ift isoneofthepointsofdiscontinuitytheinfinite
series will converge tothe mean of the values toitsleft and right,that is0.
Toobtainafeelforwhatishappening considerFigure23.13. Graphs(a),(b)and(c)
showtheeffectofincludingmoreandmoretermsintheseries.Asmoretermsaretaken
f(t)
p
–p
p
3p
t
Figure23.12
Sawtooth waveform.