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Does the following notation just mean that the integral isn't positively infinite? $$\int _a^b f(x)\,\mathrm{d}x < \infty$$

Does it not also mean that the integral converges to some finite value?

Does it not also mean that the integral doesn't diverge to (minus) infinity? Or, do I, for the last purpose, have to emphasize by writing

$$\left|\int _a^b f(x)\,\mathrm{d}x\right| < \infty$$

gebruiker
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kalkanistovinko
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  • I don't see that there is meaning to saying a value is less than infinity. Certainly, without the absolute value, it could be $-\infty$ (that would satisfy the inequality). – Jared Sep 26 '14 at 08:15
  • Well, I am used to mathematical notation occurring in the physics literature. Maybe mathematicians aren't used to this notation? – kalkanistovinko Sep 26 '14 at 08:19
  • Here is an example: $\lim_{b\rightarrow \infty}\int_1^b \ln(x)dx$ and $\lim_{b\rightarrow \infty}\int_1^b x$. Both integrals tend towards infinity, but we can definitely say that $\int_1^b \ln(x) < \int_1^b x$. So it would appear that $\lim_{b \rightarrow \infty}\left( \int_1^b \ln(x) < \int_1^b x\right)$ and thus $\lim_{b\rightarrow \infty}\int_1^b \ln(x) < \infty$. This is because an infinite value can be less than a "larger" infinite value. – Jared Sep 26 '14 at 08:20
  • Actually when a new notation is introduced you cannot judge it according to an old one, whenever the distinction is clear. For example see how mathematicians use the equality sign with the O notation, which really doesn't mean an equality of both sides. But it's understood in its context with its new definition. – kalkanistovinko Sep 26 '14 at 08:24
  • You can introduce any notation you like as long as you define it. If you are defining $\left|\int_a^b f(x)dx\right| < \infty$ to mean that the integral has a finite value, then that is what it means and you aren't really asking a question other then whether or not others adhere to this notation. – Jared Sep 26 '14 at 08:30
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    Jared there's a flaw in your reasoning - strict inequality is not preserved under limiting operations. For instance, $1/n < 2/n$ but in the limit as $n\to\infty$ they are equal. – pre-kidney Sep 26 '14 at 08:37
  • Dear Jared, this notation occurs frequently in physics textbooks, and I'm asking my question to ascertain the full meaning of what I'm reading. kkkk? – kalkanistovinko Sep 26 '14 at 08:39
  • @pre-kidney Your example is one where the limit exists (as some value goes to $\infty$). My example was comparing two diverging values. I'm not confident enough to say I'm correct, but I think your example doesn't necessarily apply to my case. You argue that two values which obviously are less than each other are not once you take the limit (then $\leq$ is more proper), but I am talking about comparing infinities which are not (in my opinion) comparable. – Jared Sep 26 '14 at 08:51
  • @Jared Might I suggest you turn your comment(s) into an answer. This may help to remove this question from the unanswered queue. If you do, you can post a link to your answer here to draw some attention to it and possibly get some upvotes. – gebruiker Jul 06 '17 at 12:27

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I view this as what one might call "physicist's notation", which does not have a precise definition attached but whose meaning can be inferred easily. It can be understood to mean

It is not the case that $\int_a^b f(x)\,\mathrm dx=\infty$,

or perhaps

$\int_a^b f(x)\,\mathrm dx$ has a value and that value is not $\infty$,

(because one would not say that $\int_0^\infty\cos x\,\mathrm dx<\infty$.) I have always seen it used in association with an integral that is nonnegative, so there is no need to emphasize that $\int_a^b f(x)\,\mathrm dx\ne-\infty$.

I would say it is rare to encounter an integral where you really need to convince the reader that it does not go to infinity in either direction. But if you do, it would be clearest to just state in words that $\int_a^b f(x)\,\mathrm dx$ is finite.

  • agreed. There may be rare cases where you allow value $-\infty$ but not $+\infty$; but if so, say it, don't just write this as ${} < \infty$. – GEdgar Jul 06 '17 at 13:02