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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?

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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 $F(s)$

Mohammad Riazi-Kermani
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  • But isn't your explanation of the domain and codomain different from that of the text I quoted? I agree that yours makes sense, but the explanation I quoted does not seem to make sense? It claims that the time domain $t$ "contains" the function $F(t)$. But it does not make mathematical sense to say that a domain $t$ "contains" a function? –  Jul 12 '18 at 10:49
  • The definition in the text leaves room for confusion. Words like time domain or frequency domain should have been carefully defined to avoid confusion. – Mohammad Riazi-Kermani Jul 12 '18 at 10:55
  • Ahh, yes, It seems that the phrases "time domain" and "frequency domain" have specific mathematical definitions that I was not aware of: https://en.wikipedia.org/wiki/Time_domain and https://en.wikipedia.org/wiki/Frequency_domain –  Jul 12 '18 at 11:07
  • Even so, with reference to the Wikipedia articles, it seems that the author has used the terminology incorrectly? The time domain is not $t$ as the author implies but, rather, $F(t)$; the frequency domain is not $s$ as the author implies but, rather, $\mathcal { F(t) }$? –  Jul 12 '18 at 11:10
  • Correct. The time domain is ${f(t)}$ The frequency domain is$ {F(s)}$ – Mohammad Riazi-Kermani Jul 12 '18 at 11:37
  • Yes, sorry, I meant $\mathcal{L} { F(t) }$. Thanks for the help. –  Jul 12 '18 at 11:39
  • Thanks for your attention to details. – Mohammad Riazi-Kermani Jul 12 '18 at 11:40