What is the optimal value for c when calculating the inverse Laplace transform?

[eljose79]

Let,s suppose we want to get the inverse Laplace transform of a function f(s) numerically,we should calculate the integral from (c-i8,c+i8) of exp(st)f(s) my question is what c we should choose for calculating the integral?..wouldn,t depend the integral of the value of c..where could i find the proof that the Inverse Laplace transform does not depend on the value of c chosen?..

Another question,let,s take the Laplace transform in 2 dimensions hten how would we define the Laplace inverse transform?..

 

[Feynman]

you can use the complexe analysis and résidue theorem

 

[eljose79]

but what would happen if i try solving it numerically for example we should calculate the integral over all R of exp(ixt)f(c+ix)exp(ct) and this would be equal to our inverse Laplace transform, the problem is what c would i choose?..thanx.

 

[Feynman]

So you have a complex integral (the integral of the inverse Laplace is a line integral in complex analysis) .
It is impossible to calculate numerically these integral , because it is a line complex integral and not a simple integral like the reel integral , for this (if you know the complex analysis) their is an important theorem to calculate the value of the complex integral it is : Residue theorem, so use this theorem

 

[eljose79]

But making the change of variable c+iu the integral becomes simply a Fourier inverse transform (is a integral on the real plane of exp(iu)f(c+iu)exp(ct)) so if we can have a real integral and should be able to compute it numerically.

 

[Feynman]

It is wrong
When you change the variale c+iu=v , so u change an COMPLEX variable and not a real variable!

 

[eljose79]

but the Laplace inverse transform is not just a special case of Fourier transform?

 

[Feynman]

Ya
the Laplace inverse transform is not just a special case of Fourier transform
But when we calculate the Laplace we use the complexe variables and not the real variables