Unformatted text preview: f ( x ) = 0 if x <2 , 1 if2 < x <1 , 2 if1 < x < 1 , 3 if 1 < x < 3 , 0 if 3 < x. 17. We use the deFnition (7) of the derivative of a generalized function and the fact that the integral against a delta function δ a picks up the value of the function at a . Thus ² φ ± ( x ) , f ( x ) ³ = ² φ ( x ) ,f ± ( x ) ³ =² φ ( x ) , f ± ( x ) ³ =² δ ( x )δ 1 ( x ) , f ± ( x ) ³ =f ± (0) + f ± (1) . 21. ±rom Exercise 7, we have φ ± ( x ) = 1 a ( U2 a ( x ) Ua ( x ) )1 a ( U a ( x ) U 2 a ( x ) ) . Using (9) (or arguing using jumps on the graph), we Fnd φ ±± ( x ) = 1 a ( δ2 a ( x )δa ( x ) )1 a ( δ a ( x )δ 2 a ( x ) ) = 1 a ( δ2 a ( x )δa ( x )δ a ( x )+ δ 2 a ( x ) ) . 25. Using the deFnition of φ and the deFnition of a derivative of a generalized...
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 Fall '11
 StuartChalk
 Derivative, Fourier Series, Dirac delta function

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