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MA22S3 Outline Solutions Tutorial Sheet 4.12 28 October 2009 Questions 1. (4) Express the following periodic function (L = 2π) as complex Fourier series 0 −π < t < −a 1 −a < t < a f (t) = 0 a<t<π (1) where a ∈ (0, π) is a constant. P Solution: f (x) = n∈Z cn eint with Z π Z a 1 1 −int dt e f (t) = dt e−int cn = 2π −π 2π −a so that c0 = a/π and a 1 e−int 1 eian − e−ian 1 cn = = = sin an. 2π −in −a πn 2i πn 2. (4) Compute the Fourier transform of f (t) = e−a|t| where a is a positive constant. Solution: The idea here is to plug f (t) directly in to the equation for f˜(k) and then use |t| = t for t > 0 and |t| = −t for t < 0. Z ∞ 1 ˜ f (k) = dt e−ikt f (t) 2π −∞ Z ∞ Z ∞ Z 0 1 1 −ikt −a|t| −ikt−at −ikt+at = dt e e = dt e + dt e 2π "−∞ 2π 0 # −∞ ∞ 0 e−t(a+ik) et(a−ik) 1 − − = 2π a + ik 0 a − ik −∞ 1 1 1 a 1 = + . = 2π a + ik a − ik π a2 + k 2 f can be represented as a Fourier integral Z ∞ Z a ∞ eikt ikt ˜ f (t) = dk e f(k) = . dk 2 π −∞ a + k2 −∞ 1 2 Conor Houghton, houghton@maths.tcd.ie, see also http://www.maths.tcd.ie/~houghton/MA22S3 Including material from Chris Ford, to whom many thanks. 1