By Baker S. G.

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2 Laplace Transform Pairs for Common Functions f (t) 1 u0 ( t ) 1⁄s 2 t u0 ( t ) 1⁄s 3 4 5 6 7 2−22 F( s) n 2 n! t u0 ( t ) ----------n+1 s δ(t) 1 δ(t – a) e e – at u0 ( t ) n – at t e u0 ( t ) – as 1---------s+a n! 4 The Laplace Transform of Common Waveforms In this section, we will present procedures for deriving the Laplace transform of common waveforms using the transform pairs of Tables 1 and 2. 5 below. 1. 1. Waveform for a pulse We first express the given waveform as a sum of unit step functions as we’ve learned in Chapter 1.

13) Now, we let t – a = τ ; then, t = τ + a and dt = dτ . 3 Frequency Shifting Property The frequency shifting property states that if we multiply a time domain function f ( t ) by an expo– at nential function e where a is an arbitrary positive constant, this multiplication will produce a shift of the s variable in the complex frequency domain by a units. 14) Proof: L {e – at f( t) } = ∞ ∫0 – st dt = ∞ ∫0 f ( t ) e – ( s + a )t dt = F ( s + a ) Note 2: A change of scale is represented by multiplication of the time variable t by a positive scaling factor a .

1 jωt – jωt We can use the relation cos ωt = --- ( e + e ) and the linearity property, as in the derivation of the transform of 2 − d sin ω t on the footnote of the previous page. 16). 66) for σ > 0 and a > 0 . 2. 2 Laplace Transform Pairs for Common Functions f (t) 1 u0 ( t ) 1⁄s 2 t u0 ( t ) 1⁄s 3 4 5 6 7 2−22 F( s) n 2 n! t u0 ( t ) ----------n+1 s δ(t) 1 δ(t – a) e e – at u0 ( t ) n – at t e u0 ( t ) – as 1---------s+a n! 4 The Laplace Transform of Common Waveforms In this section, we will present procedures for deriving the Laplace transform of common waveforms using the transform pairs of Tables 1 and 2.