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Differential Equations by Pearson
book

Differential Equations by Pearson

by E. Rukmangadachari
May 2024
Intermediate to advanced content levelIntermediate to advanced
472 pages
13h 25m
English
Pearson India
Content preview from Differential Equations by Pearson
7-32 Diff erential Equations
Proof Let g(x) be the inverse Fourier transform of G(s) so that
1
() ()
2
isx
gx Gse dx
−∞
=
p
(a) Taking conjugates on both sides
(
)
1
() ()
2
1
()() () ()
2
1
() ()
2
1
() () ,
2
(Changing the order of integration)
isx
isx
isx
gx Gse ds
fxgxdx fx Gse dsdx
Gs f xe dx ds
Gs f s ds
−∞
∞∞∞
−∞ −∞ −∞
∞∞
−∞ −∞
−∞
=
⎛⎞
∴=
⎜⎟
⎝⎠
=
=
∫∫∫
∫∫
p
p
p
p
by the definition of Fourier transform
(b) Setting g(x) = f ( x) for all x we obtain
22
1
() () ( ) ()
2
1
() ()
2
FsFsds f xf x dx
F s ds f x dx
∞∞
−∞ −∞
∞∞
−∞ −∞
=
⇒=
∫∫
∫∫
p
p
7.10 PARSEVAL’S IDENTITIES FOR FOURIER SINE
AND COSINE TRANSFORMS
Theorem 7.10.1 Similarly, we can obtain the follow ing results:
1.
00
2
() () ( ) ( )
ss
F
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Publisher Resources

ISBN: 9781299487406Publisher Website