Math450hw5sol

# Math450hw5sol - MATH 450 Section 002 Homework 5 Solutions...

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Unformatted text preview: MATH 450 Section 002 Homework 5 Solutions Winter 2009 Section 17.9, p. 919, #2. (b) a ( ω ) = 1 π Z ∞-∞ f ( x ) cos( xω )d x = 1 π Z L x cos( xω )d x = 1 π x sin( xω ) ω L x =0- 1 π Z L sin( xω ) ω d x = L π sin( Lω ) ω + 1 π cos( xω ) ω 2 L x =0 = L π sin( Lω ) ω + 1 π cos( Lω ) ω 2- 1 ω 2 = Lω sin( Lω ) + cos( Lω )- 1 πω 2 . b ( ω ) = 1 π Z ∞-∞ f ( x ) sin( xω )d x = 1 π Z L x sin( xω )d x =- 1 π x cos( xω ) ω L x =0 + 1 π Z L cos( xω ) ω d x =- L π cos( Lω ) ω + 1 π sin( xω ) ω 2 L x =0 =- L π cos( Lω ) ω + 1 π sin( Lω ) ω 2 = sin( Lω )- Lω cos( Lω ) πω 2 . So the Fourier integral representation of f is f ( x ) = Z ∞ Lω sin( Lω ) + cos( Lω )- 1 πω 2 cos( xω ) + sin( Lω )- Lω cos( Lω ) πω 2 sin( xω ) d ω. 1 (g) Integration by parts twice yields, Z ∞ e- x cos( xω )d x = (- e- x cos( xω ) ) ∞- Z ∞ e- x ω sin( xω )d x = 1- (- e- x ω sin( xω ) ) ∞- Z ∞ e- x ω 2 cos( xω )d x = 1- ω 2 Z ∞ e- x cos( xω )d x. Thus a ( ω ) = 1 π Z ∞ e- x cos( xω )d x = 1 π (1 + ω 2 ) . The calculation above gives us also Z ∞ e- x sin( xω )d x = 1 ω 1- Z ∞ e- x cos( xω )d x = 1 ω- 1 ω (1 + ω 2 ) = ω 2 ω (1 + ω 2 ) = ω 1 + ω 2 . Thus b ( ω ) = 1 π Z ∞ e- x sin( xω )d x = ω π (1 + ω 2 ) . So we have f ( x ) = Z ∞ 1 π (1 + ω 2 ) cos( xω ) + ω π (1 + ω 2 ) sin( xω ) d ω. Section 17.10, p. 932, #6 (a). 2 F{ 4 x 2 e- 3 | x | } = 4 F{ x 2 e- 3 | x | } = (17) 4 i 2 d 2 dω 2 F{ e- 3 | x | } = (4)- 4 d 2 dω 2 6 ω 2 + 9 . (m) F- 1 { 1 ω 2 + iω + 2 } = F- 1 { 1 ω- i 1 ω + 2 i } = F- 1 { i 1 + iω- i 2- iω } = (21) F- 1 { i 1 + iω }F- 1 {- i 2- iω } = (2)+(3) H ( x ) e- x ? H (- x ) e 2 x . Section 17.10, p. 932, #12. (a) Fourier transform gives ( iω ) 2 ˆ u- α 2 ˆ u =- ˆ f. Thus ˆ u ( ω ) = 1 ω 2 + α 2 ˆ f ( ω ) . The convolution theorem implies now that u ( x ) = 1 2 | α | e-| α || x | ? f ( x ) ....
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## This note was uploaded on 04/24/2010 for the course MATH 425 taught by Professor K during the Spring '10 term at University of Michigan-Dearborn.

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Math450hw5sol - MATH 450 Section 002 Homework 5 Solutions...

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