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231 Tutorial Sheet 13.12 3 Febuary 2006 Useful facts: • The Fourier integral or Fourier transform: Z ∞ dk fg (k)eikx f (x) = −∞ Z ∞ 1 g f (k) = dx f (x)e−ikx 2π −∞ • The Dirac delta function: Z ∞ dx f (x)δ(x) = f (0) −∞ • If h(x) is a continuous function Z ∞ dx f (x)δ[h(x)] = −∞ X xi :f (xi )=0 f (xi ) |h0 (xi )| Questions 1. Express the following functions as Fourier integrals: (a) f (x) = cos x |x| < 0 |x| > π 2 π 2 (b) f (x) = sin x x 2. Prove the following properties of the Fourier transform (a) The Fourier transform of an even function is even. (b) The Fourier transform of a real odd function is purely (c) f˜0 (k) = ik f˜(k). (d) Acting with the Fourier transform four times reproduces the original function apart from an overall constant. 1 2 Conor Houghton, [email protected], see also http://www.maths.tcd.ie/~houghton/231 Including material from Chris Ford, to whom many thanks. 1 3. Compute (a) (b) (c) (d) (e) Z ∞ dx x2 δ(x − 3) −∞ Z Z Z ∞ dx δ(x2 + x) −∞ 2 dx ex δ 0 (x − 1) 0 ∞ dx e−ax δ(cos x) 0 Z ∞ dx δ(eax cos x). 0 (f) d aθ(x) e . dx where a is a constant. 2