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

746 Chapter 23 Fourier series
y
y
0 T –2
t
0 T –2
t
Figure23.20
Function defined over interval0 <t < T 2 .
Figure23.21
An even periodic extension.
y
y
0 T –2
t
0 T –2
t
Figure23.22
An odd periodicextension.
Figure23.23
Aperiodic extension that is neither even nor odd.
but we have now achieved our objective of finding a periodic function. We can find
the Fourier series of this periodic function and within the interval of interest this will
converge to the required function. What happens outside this interval is not important.
Moreover, since the periodic function is even the Fourier series will contain no sine
terms.
An alternative periodic extension is that shown in Figure 23.22, which has been obtainedbyreflectinginboththeverticalandt
axesbeforerepeatingitperiodicallytogive
a periodic odd function. Its Fourier series will contain no cosine terms and within the
interval of interest will converge tothe function required.
AthirdalternativeperiodicextensionisshowninFigure23.23.However,thisextensionisneitheroddnorevenandsoithasnoneofthedesirablepropertiesoftheothertwo.
Whicheverextensionwechoose,theresultingFourierseriesonlygivesarepresentation
of the original function in the interval 0 < t < T and as such is termed a half-range
2
Fourier series. Similarly we have the terminology half-range sine series for a series
containingonlysinetermsandhalf-rangecosineseriesforaseriescontainingonlycosine
terms. The Fourier series formulae then simplify to give the following half-range
formulae:
Half-range sineseries:
a 0
= 0,a n
= 0
b n
= 4 T
∫ T/2
0
fornapositive integer
f(t)sin 2nπt
T
and f (t)isgiven by
∞∑
f(t)= b n
sin 2nπt
T
n=1
dt fornapositive integer (23.7)