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Laplace Properties and Transforms

Module by: Thanos Antoulas, JP Slavinsky. E-mail the authors

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Summary: Properties of Laplace Transforms

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Laplace Properties

ℒft=Fs=∫0-∞ftⅇ-stdt ℒ f t F s t 0- f t s t (1)
Figure 1
Property Time-domain Frequency-domain
Linearity af1t+bf2t a f1 t b f2 t aF1t+bF2t a F1 t b F2 t
Shifting in ss-domain ⅇs0tft s0 t f t Fs−s0 F s s0
Time Scaling (a>0a0) fat f a t 1−aFs−a 1 a F s a
Convolution (causal functions) f1t*f2t f1 t f2 t F1sF2s F1 s F2 s
Differentiation in Time ddtft t1 f t sFs−f0- s F s f 0-
Differentiation in Freq. -tft t f t ddsFs s1 F s
Integration in Time ∫0-tfτdτ τ 0- t f τ 1−sFs 1 s F s

Unilateral Laplace Transforms

Note: ItIt is a step function.

Figure 2
Time-domain Frequency-domain
δt δ t 11
It I t 1s 1 s
tIt t I t 1s2 1 s 2
tnIt t n I t n!sn+1 n s n 1
ⅇatIt a t I t 1s−a 1 s a
tⅇatIt t a t I t 1s−a2 1 s a 2
cosatIt a t I t ss2+a2 s s 2 a 2
sinatIt a t I t as2+a2 a s 2 a 2

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