How to find the Laplace transform of f(t±a)?

Beckett Henry 2022-09-16 Answered
How to find the Laplace transform of f ( t ± a )?
I am trying to find the laplace transform of f ( t ± a ). I did the integration myself and stopped at this point
e ± a s [ ± a 0 e s k f ( k ) d k + 0 e s k f ( k ) d k ]
which gives rise to this term
e ± a s [ ± a 0 e s k f ( k ) d k + F ( s ) ]
Where L [ f ( t ) ] = F ( s ) and t ± a = k
I couldn't figure out the value of this term
± a 0 e s k f ( k ) d k
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Answers (2)

Peugeota2p
Answered 2022-09-17 Author has 14 answers
Heaviside step function, since for t a f ( t a ) = 0
L { f ( t a ) u ( t a ) } = 0 f ( t a ) u ( t a ) e s t d t
The Heaviside step function, changes the bound of the integral:
L { f ( t a ) u ( t a ) } = a f ( t a ) e s t d t
Substitute t a = u d u = d t and the bounds change ( for a not at infinity) in the integral t = a u = 0:
L { f ( t a ) u ( t a ) } = 0 f ( u ) e s ( u + a ) d u
L { f ( t a ) u ( t a ) } = e a s 0 f ( u ) e s u d u
Finally:
L { f ( t a ) u ( t a ) } = e a s F ( s )

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Hrefnui9
Answered 2022-09-18 Author has 2 answers
Just use the definition.
0 + e t s f ( t ± a )   d t
Using y = t ± a we easily find
0 + e s ( y ± a ) f ( y )   d y = e ± s a f ^ ( y )
This is a well known property, called frequency shifting.

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