Laplace Transforms: Time Domain $t$, Frequency Domain $s$, and the Functions $F(t)$ and $\mathcal{L} \{ F(t) \}$
I read the following about Laplace transforms:
The time domain $t$ will contain all those functions $F(t)$ whose Laplace transform exists, whereas the frequency domain $s$ contains all the images $\mathcal{L} \{ F(t) \}$.
I find this explanation suspect. The time domain is $t$ -- so how does it make sense to say that it "contains" the function $F(t)$? I don't understand how it makes mathematical sense to say that a domain $t$ "contains" a function $F(t)$? My understanding is that $F(t)$ would be the codomain, whereas $t$ would be the domain (time domain). And I would argue the same with the frequency domain $s$ and $\mathcal \{ F(t) \}$.
Can someone please clear this up?
$\endgroup$1 Answer
$\begingroup$As you know the Laplace transform is an Operator which transforms a function $f(t)$ into another function $$F(s) =\mathcal{L} \{ f(t) \}$$
We can think of $f(t)$ as the input and $F(s)$ as the output.
Thus the domain of this operator is the set of all functions $f(t)$ for which $$F(s) =\mathcal{L} \{ f(t) \}$$ is convergent and the range or codomain is the set of those well defined transforms $F(s)$
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