Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN: 9781133382119
Author: Swokowski
Publisher: Cengage
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 = Fc(w):
S[f](w) = Fs(W)
=
2
πT
√ f(x) cos(wx) dx,
2
=
f(x) sin(wx) dx.
a) For any a > 0, we have the following.
C[f(ax)](w) = — Fc (~)
α
1
S[f(ax)](w) = ±F, (~2)
α
b)
ō
d
C[xf](w) ==
dw
F,(w)
d
S[xf](w) =
-Fc(w)
dw
C[e](w)
S[e](w)
2
1.
=
π 1+w2
2 w
π 1+w2](https://content.bartleby.com/qna-images/question/b9ca0fe0-8326-4567-9bb6-5d60d0a741b4/7d3775db-cbce-4fee-aa75-492c838d2077/fple3kq_thumbnail.png)
Transcribed Image Text:Prove in detail the following properties of the Fourier Sine and Cosine
Transforms. These transforms are given by
C[f](w) = Fc(w):
S[f](w) = Fs(W)
=
2
πT
√ f(x) cos(wx) dx,
2
=
f(x) sin(wx) dx.
a) For any a > 0, we have the following.
C[f(ax)](w) = — Fc (~)
α
1
S[f(ax)](w) = ±F, (~2)
α
b)
ō
d
C[xf](w) ==
dw
F,(w)
d
S[xf](w) =
-Fc(w)
dw
C[e](w)
S[e](w)
2
1.
=
π 1+w2
2 w
π 1+w2
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