math212,homework14January18January

math212,homework14January18January - f ( x ) and u t ( x,...

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Math 212 , Fall 2007 Instructor : Friedemann Brock Homework assignment, 14 January 2008 – 18 January 2008 1. (5 points) Show F h e - a | x | i = 2 a ( ξ 2 + a 2 ) - 1 . Hint : Use F h ( x 2 + a 2 ) - 1 i = π a e - a | ξ | , (1) and the Fourier inversion formula. 2. Use the Plancherel Theorem, D ˆ f, ˆ g E = 2 π h f, g i , ( | f | , | f | 2 , | g | , | g | 2 ∈ F (IR)), to show ( a ) (10 points) Z t - 2 sin( at ) sin( bt ) dt = π min { a ; b } , ( b ) (10 points) Z t 2 dt ( t 2 + a 2 )( t 2 + b 2 ) = π a + b . Hint : In (a) , use the formula ˆ χ a ( ξ ) = 2 ξ - 1 sin( ) , and in (b) , formula (1). 3. Consider the wave equation u tt = c 2 u xx , with initial conditions u ( x, 0) =
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Unformatted text preview: f ( x ) and u t ( x, 0) = g ( x ). (a) (10 points) Assuming that all the Fourier transforms in question exist, show that ˆ u ( ξ, t ) = ˆ f ( ξ ) cos ctξ + ˆ g ( ξ )( cξ )-1 sin ctξ. (b) (10 points) Invert the Fourier transform to obtain d’Alembert’s formula of the solution u , u ( x, t ) = 1 2 ( f ( x-ct ) + f ( x + ct )) + 1 2 c Z x + ct x-ct g ( y ) dy....
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