Consider the Laplace transform bar(f)(s) of a function f(t) defined as bar(f)(s)=int_0^(oo)f(t)e^(-st)dt Can this relation be inverted to obtain f(t) in terms of bar(f)(s)?

wasangagac4

wasangagac4

Answered question

2022-11-01

Consider the Laplace transform f ~ ( s ) of a function f(t) defined as
f ~ ( s ) = 0 f ( t ) e s t d t .
Can this relation be inverted to obtain f(t) in terms of f ~ ( s )?
The reason I ask this is as follows. The Fourier transform
f ~ ( k ) = f ( x ) e i k x d x
is defined simulataneously with the inversion formula
f ( x ) = f ~ ( k ) e i k x d k .
But I haven't seen the same for the Laplace's transform.

Answer & Explanation

Besagnoe9

Besagnoe9

Beginner2022-11-02Added 9 answers

Yes, the Laplace transform can be inverted. Based on your earlier question I suspect you're actually curious whether the inverse is given by
f ( t ) = 0 f ~ ( s ) e s t d s .
No, that is absolutely not how you invert the Laplace transform.
Bodonimhk

Bodonimhk

Beginner2022-11-03Added 3 answers

If f ~ converges on γ + i R with γ R , the inverse can be obtained as
f ( t ) = 1 2 π i lim T γ i T γ + i T f ~ ( s ) e s t d s .

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