Mathematical Methods for Physics and Engineering - Matematica.NET

INTEGRAL TRANSFORMS(i) Differentiation:F [ f ′ (t) ] = iω˜f(ω). (13.28)This may be extended to higher derivatives, so thatandsoon.(ii) Integration:F [ f ′′ (t) ] = iωF [ f ′ (t) ] = −ω 2˜f(ω),t]F[∫f(s) ds = 1iω ˜f(ω)+2πcδ(ω), (13.29)where the term 2πcδ(ω) represents the Fourier transform of the constantof integration associated with the indefinite integral.(iii) Scaling:F[f(at)] = 1 ( ω). (13.30)a˜fa(iv) Translation:(v) Exponential multiplication:where α may be real, imaginary or complex.F[f(t + a)] = e iaω˜f(ω). (13.31)F [ e αt f(t) ] = ˜f(ω + iα), (13.32)◮Prove relation (13.28).Calculating the Fourier transform of f ′ (t) directly, we obtainF [ f ′ (t) ] = √ 1 ∫ ∞f ′ (t) e −iωt dt2π −∞= √ 1 [e −iωt f(t) + 1 ∫ ∞√ iω e −iωt f(t) dt2π 2π] ∞−∞= iω˜f(ω),if f(t) → 0att = ±∞, asitmustsince ∫ ∞|f(t)| dt is finite. ◭−∞To illustrate a use and also a proof of (13.32), let us consider an amplitudemodulatedradio wave. Suppose a message to be broadcast is represented by f(t).The message can be added electronically to a constant signal a of magnitudesuch that a + f(t) is never negative, and then the sum can be used to modulatethe amplitude of a carrier signal of frequency ω c . Using a complex exponentialnotation, the transmitted amplitude is now−∞g(t) =A [a + f(t)] e iωct . (13.33)444