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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
7-12 Diff erential Equations
7.8.4 Modulation Theorem
Theorem 7.8.5 If F (s) is the complex Fourier transform of f (x) then
(a)
1
{()cos}[()()]
2
Ffx ax Fs a Fs a=++−
and
(b)
1
{()sin}[()()].
2
Ffx ax Fs a Fs a=+−−
(7.49)
Proof It is given that
() ()
isx
Fs f xe dx
−∞
=
(a) By definition
() ()
{ ( )cos } ( )cos
()
2
1
() ()
2
1
[( ) ( )]
2
isx
iax iax
isx
ix s a ix s a
Ffxax fxaxedx
ee
fx e dx
fxe dx fxe dx
Fs a Fs a
−∞
−∞
∞∞
+−
−∞ −∞
=
+
⎛⎞
=
⎜⎟
⎝⎠
⎡⎤
=+
⎢⎥
⎣⎦
=++−
∫∫
(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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Publisher Resources

ISBN: 9781299487406Publisher Website