# HW9 - f that satisfy each of the following inhomogeneous...

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Homework #9 Due: Tuesday, April 28, 2009 Note : Use of Matlab (or any other software) is not permitted. I. (Exercise 7.27) Find all generalized functions f that satisfy each of the following inhomogeneous equations. 1. (1pt) ) ( ) ( ) 1 ( 4 x x f x δ = ; 2. (1pt) ) ( ) ( ) 1 ( 4 x x f x = + ; Hint: Study exercises 7.23, 7.24. II. (Exercise 7.33, 7.36) Find the Fourier transform of each of the following generalized functions. In so doing, freely use the rules from the Fourier transform calculus together with the Fourier transforms we deduced in class. 3 . (1pt) ) ( ) ( 2 x e x f ix π = 4 . (1pt) ) cos( ) ( x x f = 5 . (1pt) ) ( ' ) ( x x x f = 6 . (1pt) ) ( ) ( 4 x sign e x f ix = 7. (1pt) 4 1 ) ( 2 = x x f III. (Exercise 7.27) Find all generalized functions
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Unformatted text preview: f that satisfy each of the following inhomogeneous equations. 8. (1pt) ) ( ) ( ) 1 ( 4 x x f D = ; 9. (1pt) ) ( ) ( ) 1 ( 4 x x f D = + ; Here D denotes the derivative operator. Thus ) ( ) ( 4 4 4 x dx f d x f D = . Hint: Fourier transform the equation, solve the algebraic equation thus obtained, and then Fourier invert the solution. IV. Find all generalized functions f that satisfy: 10 . (1pt) ) sin( ) ( ) 1 ( 2 x x f D = ; Hint: Use convolution with the Greens function, or Fourier transform the equation. Total : 10 pts (1 point each)...
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