# Connexions

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# CTFT Development 3

Module by: Richard Baraniuk. E-mail the author

The forward and inverse Laplace transforms are defined as:

Fs=fte(st)d t F s t f t s t
(1)
ft=12πiaia+iFsestd s f t 1 2 s a a F s s t
(2)
substitute in s=iω s ω and let a=0 a 0 (integrate along iω ω axis in complex plane) and we have:
Fiω=fte(iωt)d t F ω t f t ω t
(3)
ft=12πFiωeiωtd ω f t 1 2 ω F ω ω t
(4)
In other words
Fiω=Fs| s=iω F ω s ω F s
(5)
which explains the " iω ω " notation.

## Example 1

fL2R f L 2 but not LR L .

|ft|d t = t f t and |ft|2d t < t f t 2 .

## Example 2

fLR f L but not L2R L 2 .

|δt|d t =1 t δ t 1 and |δt|2d t = t δ t 2 .

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