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A textbook has the following explanation for the Laplace transform:

$f(s) = \int^\infty_0 F(t)e^{-st} \ dt$

[...]

Another way of looking at the Laplace transform is as a mapping from points in the $t$ domain to points in the $s$ domain. [...]

The time domain $t$ will contain all those functions $F(t)$ whose Laplace transform exists, whereas the frequency domain $s$ contains all the images $f(s) = \mathcal{L} \{ F(t) \}$.

In my previous question, we seemed to have concluded that there is something incorrect about this explanation; specifically, $t$ and $s$ are used twice: once as bound variables to denote the domains of $t \mapsto F(t)$ and $s \mapsto f(s)$, respectively, and a second time to denote the "time domain" and "frequency domain", respectively. It is not $t$ and $s$ that are the time and frequency domains, respectively, but, rather, $F(t)$ and $\mathcal{L} \{ F(t) \}$ that are the time and frequency domains, respectively. $t$ is the domain of $F(t)$, but this doesn't make it the "time domain"; analogously, $s$ is the domain of $f(s) = \mathcal{L} \{ F(t) \}$, but that doesn't make it the "frequency domain". So, unless we reached incorrect conclusions in my last question, it seems that the use/associations of "time domain" and "frequency domain" in this section of the text are incorrect.

I now want to bring this to the attention of the author, but I've never studied Laplace transforms, which is why I started reading this textbook in the first place. As such, I don't want to send an email that sounds incoherent, so I'd appreciate it if those who already understand Laplace transforms could help me phrase the issue with this explanation in a way that is coherent and would be likely to be well-understood by the author of the text.

Thank you.

EDIT:

The image in this question clarifies my misunderstanding of what time domain and frequency domain mean.

The time $t$ and the time domain are two separate concepts; analogously, the frequency $s$ and the frequency domain are also two separate concepts.

What made this so confusing is the fact that the author used the same variable $t$ to denote both the time $t \in (0, \infty)$ and the time domain. The same is true for the author's use of $s$ with frequency domain.

The time domain and frequency domain are not actually domains of mathematical functions; rather, the word "domain" is used colloquially in this context.

The following image makes this clear:

enter image description here

However, that still does not excuse the sloppiness of using the same variable for two different concepts and within the same context! If the author did not undertake such bad use of notation, then this entire confusion could have been avoided!

When you read the excerpt knowing all of this, it suddenly makes sense!

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1 Answers1

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Neither $t$ nor $s$ are domains. They are variables. A domain is a set of allowed values for a variable. So the '$t$ domain' is $(0,\infty),$ as indicated by the limits on the integral. The '$s$ domain' is technically all complex numbers.

Using the label '$t$' as a variable, all by itself, and then using the label '$t$' in the expression '$t$ domain' is not equivocation or incorrect usage, any more than saying the category 'bear' is not the same as the category 'polar bear'. While I've used the term 'bear' in both expressions, the term 'polar bear' is really atomic and cannot be divided. In the same way, the phrase '$t$ domain' is atomic. The '$t$' in the phrase '$t$ domain' cannot stand by itself.

In your previous question to which you linked, you wrote, "The time domain is $t$..." As I have hopefully made clear, the time domain is not $t$. The time domain is $(0,\infty)$. The variable $t$ must live in the time domain. That is, $t\in(0,\infty)$.

Does this answer your question?

Adrian Keister
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  • @Wyuw: Yes, I think it's fair to say that. The author's wording there is a tad sloppy. – Adrian Keister Jul 12 '18 at 13:59
  • Sorry, I had to delete my comment to reword it. I understand what you're saying, but the author says that "The time domain $t$ will contain all those functions $F(t)$ ...". If what you say is true, then doesn't this imply that the author's explanation is incorrect? As you say, the time domain is $(0,\infty)$, and we have that $t\in(0,\infty)$. The author is saying that the time domain is $t$ and that it contains all those functions $F(t)$. [...] –  Jul 12 '18 at 14:02
  • The time domain is a set of $t$ values. Fundamentally, the functions $F(t)$ are sets of ordered pairs $(t,F(t))$. These ordered pairs are not contained in the time domain, so the author is incorrect on that point. In other words, you not understanding is the correct reaction! – Adrian Keister Jul 12 '18 at 14:05
  • [...] But now the author has used $t$ twice in the same context: once to say that the domain of $F(t)$ is $(0,\infty)$, so that we have, as you say, $t\in(0,\infty)$, and a second time within the same context to say that $t$ is the time domain. –  Jul 12 '18 at 14:06
  • @Wyuw: Also technically incorrect, but not as liable to create confusion as the other error. – Adrian Keister Jul 12 '18 at 14:07
  • See my edit. I think this comes down to (1) bad use of notation on the author's part and (2) a misunderstanding of what "time domain" and "frequency domain" actually are. –  Jul 12 '18 at 15:19