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Laplace Properties and Transforms

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

Summary: Properties of Laplace Transforms

Laplace Properties

ft=Fs=0-fte(st)dt 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 Fss0 F s s0
Time Scaling (a>0a0) fat f a t (1a)Fsa 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 sFsf0- 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 (1s)Fs 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 1sa 1 s a
teatIt t a t I t 1sa2 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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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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