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Mathematical Methods by Pearson
book

Mathematical Methods by Pearson

by E. Rukmangadachari
May 2024
Intermediate to advanced content levelIntermediate to advanced
437 pages
17h 41m
English
Pearson India
Content preview from Mathematical Methods by Pearson
10-34 Mathematical Methods
Proof
Let the Fourier series for f (x) in (l, l) be
0
11
1
() cos sin
2
nn
nn
nx n
fx a a b
ll
pp
∞∞
==
=+ +
∑∑
x
(10.41)
Multiplying both sides of (10.41) by f (
x) and
integrating term by term from l to l we obtain
2
0
1
1
[ ()] () ()cos
2
()sin
lll
n
lll
n
l
n
l
n
a
nx
f x dx f x dx a f x dx
l
nx
bfx dx
l
p
p
−−−
=
=
=+
+
∫∫∫
(10.42)
0
0
(, , )
1
()1,cos ,sin
()1,cos ,sin ( , , )
nn
l
l
l
nn
l
aab
nx nx
fx dx
lll
nx nx
f x dx la la lb
ll
pp
pp
⎛⎞
=
⎜⎟
⎝⎠
⎛⎞
⇒=
⎜⎟
⎝⎠
Also
(10.43)
Substituting (10.42) we have
2222
0
1
1
[()] ( )
2
l
nn
l
n
fx dx l a a b
=
⎡⎤
=++
⎢⎥
⎣⎦
(10.44)
which is known as Parseval’s formula.
Related Formulae
1. If f (x) is an even function in (l, l), (10.43)
becomes
222
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Publisher Resources

ISBN: 9781299446557Publisher Website