(b) The Fourier transform of a piecewise smooth function f(x), for which f∞ f(x)|dx exists and is finite, is given by 1 ∞ F(k) = F[f(x)] = 2πT f(x)e-dr. -ikx ∞ i. Show that for a real valued function f(x) its Fourier transform F(k) satisfies the identity F(-k) = F(k) ii. Assuming that the function f(x) is real valued and odd (f(x) = − f(x)), show that its Fourier transform F(k) = F[f(x)] is a purely imaginary function, that is, Re(F(k)) = 0. iii. Find the Fourier transform F(k) for the function f(x) = x exp(−|x|). Hint: fre X 1 x exp(ax) dx = exp(ax) ― a exp(ax) + constant.
(b) The Fourier transform of a piecewise smooth function f(x), for which f∞ f(x)|dx exists and is finite, is given by 1 ∞ F(k) = F[f(x)] = 2πT f(x)e-dr. -ikx ∞ i. Show that for a real valued function f(x) its Fourier transform F(k) satisfies the identity F(-k) = F(k) ii. Assuming that the function f(x) is real valued and odd (f(x) = − f(x)), show that its Fourier transform F(k) = F[f(x)] is a purely imaginary function, that is, Re(F(k)) = 0. iii. Find the Fourier transform F(k) for the function f(x) = x exp(−|x|). Hint: fre X 1 x exp(ax) dx = exp(ax) ― a exp(ax) + constant.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![(b) The Fourier transform of a piecewise smooth function f(x), for which f∞ f(x)|dx
exists and is finite, is given by
1
∞
F(k) = F[f(x)]
=
2πT
f(x)e-dr.
-ikx
∞
i. Show that for a real valued function f(x) its Fourier transform F(k) satisfies
the identity F(-k) = F(k)
ii. Assuming that the function f(x) is real valued and odd (f(x) = − f(x)),
show that its Fourier transform F(k) = F[f(x)] is a purely imaginary function,
that is, Re(F(k)) = 0.
iii. Find the Fourier transform F(k) for the function f(x) = x exp(−|x|).
Hint:
fre
X
1
x exp(ax) dx
=
exp(ax)
―
a
exp(ax) + constant.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F69cce4ac-4bf6-4e6b-8636-bf160e045b58%2F06e5f588-fa2c-4615-8823-128bcd16e01a%2Fxu76kl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(b) The Fourier transform of a piecewise smooth function f(x), for which f∞ f(x)|dx
exists and is finite, is given by
1
∞
F(k) = F[f(x)]
=
2πT
f(x)e-dr.
-ikx
∞
i. Show that for a real valued function f(x) its Fourier transform F(k) satisfies
the identity F(-k) = F(k)
ii. Assuming that the function f(x) is real valued and odd (f(x) = − f(x)),
show that its Fourier transform F(k) = F[f(x)] is a purely imaginary function,
that is, Re(F(k)) = 0.
iii. Find the Fourier transform F(k) for the function f(x) = x exp(−|x|).
Hint:
fre
X
1
x exp(ax) dx
=
exp(ax)
―
a
exp(ax) + constant.
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