Theorem 7.8.5 If F (s) is the complex Fourier transform of f (x) then
(a)
1
{()cos}[()()]
2
FfxaxFsaFsa=++−
and
(b)
1
{()sin}[()()].
2
FfxaxFsaFsa=+−−
(7.49)
Proof It is given that
()()
isx
Fsfxedx
∞
−∞
=
∫
(a) By definition
()()
{()cos}()cos
()
2
1
()()
2
1
[()()]
2
isx
iaxiax
isx
ixsaixsa
Ffxaxfxaxedx
ee
fxedx
fxedxfxedx
FsaFsa
∞
−∞
−
∞
−∞
∞∞
+−
−∞−∞
=
+
⎛⎞
=
⎜⎟
⎝⎠
⎡⎤
=+
⎢⎥
⎣⎦
=++−
∫
∫
∫∫
(b) can be proved similarly.
Corollary 7.8.6 If F
s
(s) and F
c
(s) are the Fourier sine transform and
the Fourier cosine transform of f (x), respectively, then
1
(a){()cos}[()()]
2
1
(b){()sin}[()()]
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