pset4 - of plane waves in the form of an integral over .)...

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Physics (PHZ) 3113 Mathematical Physics Florida Atlantic University Fall, 2010 Problem Set IV Due: Thursday, 23 September 2010 1. (Problems 7.11.6 and 8, pp. 377–378) Use Parseval’s theorem and the results of the problems indicated to sum the following series: a. X n 1 1 n 4 (see Problem 7.9.9). b. X n 1 n odd 1 n 4 (see Problem 7.9.10). 2. (Problems 7.12.3 and 7, p. 384) Find the exponential Fourier transform of f (a) ( x ) := - 1 - π < x < 0 1 0 < x < π 0 | x | > π and f (b) ( x ) := ( | x | | x | < 1 0 | x | > 1 . Use these results to write each f ( x ) as a Fourier integral. (That is, write each as a superposition
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Unformatted text preview: of plane waves in the form of an integral over .) 3. (Problems 7.12.24 and 33, pp. 385386) a. Find the exponential Fourier transform of f ( x ) := e-| x | and write the inverse transform to show that Z cos x 2 + 1 d = 2 e-| x | . b. Find the Fourier transform of 1 / (1 + x 2 ). Hint : Interchange x and in the result of the rst part. c. Verify Parsevals theorem for the Fourier transform computed in part (a)....
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