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I am trying to find some examples of functions that are "just in $L^1([a,\infty))$" for some $a>0$ that can be chosen. By this I mean the following are integrable: $1/x^p$ and $1/x(\ln x)^p$ for $p>1$ but I don't think it is possible to integrate $1/x(\ln(\ln(x))^p$ explicitly- at least Mathematica fails.

My questions are:

1) Is $1/x(\ln(\ln(x))^p$ integrable? If it is then can we keep iterating to see that $1/x(\ln(\ln(...\ln(x)))^p$ is integrable? (SOLVED)

2) What are some examples of functions that are just about integrable and if they are edited slightly they fail like above with $p=1$? I would like monotonically decreasing functions but any functions will be of interest to me.

Thanks.

(Note that I am trying to find examples to give context to my dissertation so this may count as "homework" but I can chose whether or not to include it so I didn't put the homework tag. Let me know if I should!)

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2 Answers 2

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I think the change $x = \exp(u)$ will answer your 1st question.

For your second question, try to study $x^a \cdot \ln(x)^b \cdot \exp(x)^c$.

Good luck

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  • $\begingroup$ Thanks for the suggestion. When I do that change I get $\int \frac{1}{(\ln u)^p} du$ I can see this is infinite if $p<2$ but I am not sure what happens if $p\ge 2$. $\endgroup$
    – David
    Mar 26, 2013 at 15:58
  • $\begingroup$ As I said in the comments on GEdgar's answer I realise that the question in this comment is simple but question 2 is still OK and I am interested to see peoples' ideas. $\endgroup$
    – David
    Mar 26, 2013 at 21:16
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Beginning... Show $$ \frac{1}{x(\log\log x)^p} $$ is not integrable, by comparison with $$ \frac{1}{x\log x} . $$ That is, for large enough $x$, $$ \frac{1}{x(\log\log x)^p} > \frac{1}{x\log x} . $$

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  • $\begingroup$ I tried this but failed. I might be using bounds that are not sharp enough or I might just be acting stupid but I get use lmorin's suggested substitution to get $1/(\log x)^p\ge 1/x(\log x)^{p-1}$ and then this is infinite when $p-1\le 1$ but i am not what happens otherwise. $\endgroup$
    – David
    Mar 26, 2013 at 16:44
  • $\begingroup$ Of course this should be $$\int 1/(\log x)^p dx ≥ \int 1/ x (\log x)^{p−1} dx $$ above but its too late to edit! $\endgroup$
    – David
    Mar 26, 2013 at 17:07
  • $\begingroup$ If I can show that log(x)^p is eventually concave then I will be done. Any idea on how to do that? $\endgroup$
    – David
    Mar 26, 2013 at 18:22
  • $\begingroup$ Coming back to this now I have just realised how easy this question 1 is!! Question 2 is still reasonable though I think... $\endgroup$
    – David
    Mar 26, 2013 at 21:13

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