Unformatted text preview: MEC702 Wk 12 Lect 1 Linearity and Transforms of Derivatives
Theorem. (Linearity of Fourier Cosine and Fourier Sine Transforms) 1. Fc {af + bg} = aFc {f } + bFc {g}. 2. Fs {af + bg} = aFs {f } + bFs {g}. Theorem. (Fourier Cosine and Fourier Sine Transforms of Derivatives) q 1. Fc {f 0 } = wFs {f } − 2. Fs {f 0 } = −wFc {f }. 3. Fc {f 00 } = −w2 Fc {f } − q 2 0
π f (0). 4. Fs {f 00 } = −w2 Fs {f } + q 2
π wf (0). Example. 2
π f (0). Find the Fourier cosine transform of f (x) = e−ax , a > 0. Solution. We have
f (x) = e−ax f 0 (x) = −ae−ax 00 f (x)
Fc {f 00 } and = a2 e−ax
= a2 Fc {f } f 0 (0) = −ae0 = −a. Then
r Fc {f } = a2 Fc {f } = Fc {f } = 00 2 0
f (0)
π
r
2
−w2 Fc {f } −
(−a)
π
r
2
a
.
π a2 + w2
−w Fc {f } −
2 1 Fourier Transform
We dene the complex Fourier integral
f (x) = 1
2π ˆ ∞ ˆ ∞ f (v)eiw(x−v) dv dw.
−∞ −∞ Writing the exponential function as a product of exponential forms, we have
1
f (x) = √
2π ˆ ∞ −∞ 1
√
2π ˆ ∞
f (v)e−iwv dv e−wx dw −∞ from which we have the inner integral as the Fourier transform of f (v) and the
outer integral as the inverse Fourier transform of fˆ(w).
Written v = x, we have the Fourier transform of f (x) as
1
fˆ(w) = √
2π and the inverse Fourier transform of 1
f (x) = √
2π ˆ ∞ f (x)e−iwx dx
−∞ fˆ(w) as
ˆ ∞ fˆ(w)eiwx dw
−∞ Other notations for Fourier transform:
fˆ = F {f },
Example. f = F {fˆ}. Find the Fourier transform of
(
1, x < 1
f (x) =
0, otherwise. Solution. fˆ(w) =
=
=
=
= ˆ ∞
1
√
f (x)e−iwx dx
2π −∞
ˆ −1
ˆ 1
ˆ ∞
1
−iwx
−iwx
−iwx
√
f (x)e
dx +
f (x)e
dx +
f (x)e
dx
2π −∞
−1
1
ˆ 1
1
√
(1)e−iwx dx
2π −1
−iwx 1
1
e
√
2π −iw −1
1
√ (e−iw − eiw ).
−iw 2π 2 By Euler's formula, we have eiw = cos w + i sin w and e−iw = cos w − i sin w,
then
e−iw − eiw = 2i sin w. So r
fˆ(w) = Example. 2 sin w
.
π w Find the Fourier transform of
(
f (x) = Solution. F {e −ax } 1
√
2π
1
√
2π
1
√
2π
1
√
2π =
=
=
= ˆ e−ax ,
0, x > 0, a > 0
x < 0. ∞ f (x)e−iwx dx
−∞
ˆ 0 ˆ
f (x)e −iwx −∞
∞
−ax −iwx ˆ e e ∞ dx + f (x)e −iwx
dx 0 dx 0 e−(a+iw)x
−(a + iw) ∞
=√
0 1
.
2π(a + iw) Linearity and Transform of Derivatives
(Linearity of the Fourier Transform) The Fourier transform is a
linear operation, that is,
Theorem. F {af + bg} = aF {f } + bF {g}.
Theorem. (Fourier Transform of Derivatives of 1. F {f 0 } = iwF {f }. 2. F {f 00 } = −w2 F {f }. Example. f (x)) Find the Fourier transform of f (x) = xe−x .
2 Solution. F {xe −x2 } =
=
=
=
=
1 −x2
d
− e
F
dx 2
o
2
1 n
− F e−x
2
n
o
2
1
− iwF e−x
2
1
1 −w2 /4
− iw √ e
2
2
iw −w2 /4
− √ e
.
2 2 3 ...
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 Winter '16
 Alveen Chand
 Fourier Series, Complex number, Euler's formula

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