Compute this ILT ccL^(-1){(s+1)/(z^s)} where |z|>1

wasangagac4

wasangagac4

Answered question

2022-10-16

Try to compute this ILT
L 1 { s + 1 z s } ,
where |z|>1. However, I'm not sure this is possile?

Answer & Explanation

Travis Sellers

Travis Sellers

Beginner2022-10-17Added 18 answers

I worked the actual Bromwich integral directly because the residue theorem is no help here. You get something in terms of c, the offset from the imaginary axis of the integration path. So I appealed to something more basic. Consider
f ^ ( s ) = 0 d t f ( t ) e s t
Just the plain Laplace transform of some function f. Let's throw away any conditions on continuity, etc. on f, and consider
f ( t ) = δ ( t log z )
for some z. Then
f ^ ( s ) = z s
Interesting. Now consider f ( t ) = δ ( t log z ); then
f ^ ( s ) = s z s + δ ( log z )
It follows that
L { δ ( t log z ) + δ ( t log z ) } = s + 1 z s + δ ( log z )
Therefore
L 1 { s + 1 z s } = δ ( t log z ) + δ ( t log z ) δ ( log z ) δ ( t )

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Differential Equations

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?