The only time you can transform term-by-term in that way is if you have a SUM. So, for example,

,

but things are more complicated with a PRODUCT, in particular,

.

There is NO product rule for Laplace Transforms EXCEPT for when you want to send — that is except when one of the functions you are transforming is an exponential.

When one of the functions is an exponential the First Shift(/Translation) Theorem applies. This states:

.

Let us break this down a little bit.

is the notation for the Laplace transform of , that is:

.

When the function is multiplied by an exponential in the time domain that has the effect of SHIFTING the function is the -domain.

This shifting, which is a horizontal translation in the graph, has the effect of shifting to .

In laymen terms — well inasmuch as these can be made into laymen terms:

.

– The ‘‘ just means you transform as normal

– The means that it isn’t evaluated at (e.g. ) but at (e.g. ).

So in the case of you transform the as normal… but then shift the to .

It can be hard to write this down. I suggest transform the first:

,

and then shift the :

.

Regards,

J.P.

With regards to Q. 1 (iii) can you tell me where that rule came from. I’m totally stumped and I’ve read your answer but it is still not clicking with me. Is it not a case of transforming by its own rule and by its rule?

]]>