MEC702_Wk_12_Lect_1 - MEC702 Wk 12 Lect 1 Linearity and...

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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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