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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
Fourier Integral Transforms 7-17
()
(
)
(
)
(
)
00
0
0
0
0
0
2
sin 1 sin
cos
2
sin
cos cos0
2
sin
sin 1 cos
2
sin 1 cos
,0
2
2
0,
sx st dt ds
st
sx ds
s
sx ds
s
sx s
ds
s
sx s
x
ds f x
s
x
∞∞
=⋅
⎛⎞
=
⎜⎟
⎝⎠
−+
⎛⎞
=
⎜⎟
⎝⎠
=
≤<
⇒==
>
∫∫
p
p
p
p
p
p
p
p
p
p
p
p
At x = p which is a point of discontinuity for f (x) the value of the
integral is
(0)(0) 10
22 224
ff−+ + +
⎛⎞⎛⎞
⋅==
⎜⎟⎜⎟
⎝⎠⎝⎠
pp p p p
Example 7.8.14 Find the Fourier transform of
2
1, 1
()
0, 1
xx
fx
x
−≤
=
>
Hence, evaluate
3
0
cos sin
cos .
2
xx x
x
dx
x
[JNTU 2002, 2004S]
Solution The Fourier transform of f (x) is given by
11
2
11
1
2
23
1
23
23 3
{()} () ()
0(1)0
(1 ) 2 2
() ()
1(1)
02 2
44 4
cos sin ( cos sin )
isx
isx isx isx
isx isx isx
is is is is
Ffx Fs fxedx
ed ...
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Publisher Resources

ISBN: 9781299487406Publisher Website