If we change the interval from (b,b+ 2p) of length
2p to an interval (a,a+ 2l) of length 2l then we
have to substitute
22
,
zx
l
x
zdzdx
ll
p
pp
==
⇒==
variable
length of the interval
(10.29)
Also,
z=b⇒x =
l
p
b=a (say)
z =b +2p
⇒ x
l
p
(b + 2p) =
l
p
β
+ 2l =a +2l
Now, the Fourier series of F ( z) =F
(
)
xlp
=
f (x)(say) is given b
y
0
11
()cossin
2
nn
nn
a
nxnx
fxab
ll
pp
∞∞
==
=++
∑∑
(10.30)
where the Fourier coeffi cients a
0
, a
n
and b
n
are given
by Euler’s formulas
+
⎛⎞
=
⎜⎟
⎝⎠
∫
2
0
1
(,,)()1,cos,sin
l
nn
nxnx
aabfxdx
lll
a
a
pp
(10.31)
Note 1 Putting x= 0 in (10.31) we have formu-
las for f (x) defined in (0, 2l) and putting ...
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