Proof Let g(x) be the inverse Fourier transform of G(s) so that
1
()()
2
isx
gxGsedx
∞
−
−∞
=
∫
p
(a) Taking conjugates on both sides
(
)
1
()()
2
1
()()()()
2
1
()()
2
1
()(),
2
(Changing the order of integration)
isx
isx
isx
gxGseds
fxgxdxfxGsedsdx
Gsfxedxds
Gsfsds
∞
−∞
∞∞∞
−∞−∞−∞
∞∞
−∞−∞
∞
−∞
=
⎛⎞
∴=
⎜⎟
⎝⎠
=
=
∫
∫∫∫
∫∫
∫
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
FsFsdsfxfxdx
Fsdsfxdx
∞∞
−∞−∞
∞∞
−∞−∞
=
⇒=
∫∫
∫∫
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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