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I'm trying to find the name of, and a good online reference to, a type of "logarithm series", e.g.

$$(1+x)^9 = \sum_{k=0}^{\infty} \frac{9^k\ln^k(1+x)}{k!} $$

I realise that this comes from $x^y \equiv \operatorname{e}^{y\ln x}$ and the Taylor series of the exponential function.

Wolfram-Alpha gives this as a potential series for $(1+x)^9$ alongside, for example, the binomial. I also remember reading about this type of "logarith series" some years ago. I've tried to Google it but, as you can probably imagine, it just lists the series of the natural log.

Fly by Night
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  • See here. Seems it has to do with the Hurwitz-Zeta function. – Bennett Gardiner Oct 28 '13 at 02:57
  • I think I am not clear about the question. Are you trying to find a series in logarithms for (1+x)$^9$ or are you asking a more general question? – Betty Mock Oct 28 '13 at 03:07
  • @BettyMock As my first line says: "I'm trying to find the name of, and a good online reference to, a type of logarithm series...". I think that's pretty clear. We have power series and Taylor series and Fourier series. What are these called and are they studied in depth anywhere. – Fly by Night Oct 29 '13 at 15:01
  • In that case, you've already answered you own question. The series you provided is the Taylor series of $e^{9\ln(1+x)}=(1+x)^9$. And no, your question was anything but clear. – Lucian Oct 29 '13 at 17:08
  • @Lucian No. On line three I say that I realise (this example) comes from a Taylor series. But this is just one example. The point is that it is a series of the form $\sum a_k \ln^k[\operatorname{f}(x)]$ and I would like to know more about these series. Please re-read my post. As I said: Wolfram Alpha lists this as a possible series, along side others, as though it is a different type of series. – Fly by Night Oct 29 '13 at 17:25
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    Uhm... OK. But -in case you haven't noticed- I'm not the only one who has been unable to help you, nor the only one to have expressed doubts concerning the clarity of your post. And this despite the fact that the question has even been put up for bounty for several days already, so you can't say that it has been ignored or overlooked... Perhaps something to think about ? – Lucian Oct 30 '13 at 22:58
  • @Lucian The up-votes and the two stars show that it was a perfectly valid, comprehensible and interesting question. – Fly by Night Oct 30 '13 at 23:37
  • Every series of the form $\sum a_k(g(x))^k$ can be written as $\sum a_k \log^k(f(x))$ by the simple expedient of letting $f(x)=e^{g(x)}$, so I don't see where this is really a question about series involving logarithms. – Gerry Myerson Nov 01 '13 at 23:02
  • @GerryMyerson I already made this point in my original post. I have also explained the history and motivation in my post. – Fly by Night Nov 02 '13 at 18:03
  • Well, evidently you haven't explained it well enough for anyone to understand it and to contribute anything you find useful. Maybe it's time to edit your question to clarify matters. – Gerry Myerson Nov 02 '13 at 21:57
  • @GerryMyerson Perhaps, or perhaps this is not a very well-known or well-studied topic. That would explain me, and other users, not being familiar with the set-up. – Fly by Night Nov 03 '13 at 19:11

1 Answers1

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We do have Taylor's Series and Fourier Series and each is extremely useful. There are some applications where one of them is of the most use; some where the other is; and a few where either would do.

Series in terms of powers of log(1+x) are a form of Taylor's series in which you expand f(x) in a Taylor's series, substitute log(x) for x in the series, and thus get an expansion for f(logx). Lucian has pointed this out for the function in your example.

I think the reason you are not finding good references is that series like this are not used for general purposes as are the Taylor's and Fourier series. Mathematicians develop analysis of ideas that are useful and widely applicable. Since the Taylor's series include the kind of logarthmic series you are interested in, all the work that has been done in developing that theory applies to your interest.

Betty Mock
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  • Betty - I already pointed out your middle paragraph in my original question. As for the third paragraph, as I said in my post, Wolfram Alpha uses this type of series, I remember seeing them before and I can't find references because if you put series and log in the same search you get pages on the power series for $\ln(1+x)$. The example I gave comes from Taylor series, but not all $\sum a_k\ln^k[\operatorname{f}(x)]$ do. I have mentioned all of this previously. – Fly by Night Oct 30 '13 at 11:39
  • Did you mention $ln^k[f(x)]$ previously? I can't find it. – Betty Mock Oct 31 '13 at 19:27
  • Try the fifth comment to my original post. – Fly by Night Nov 01 '13 at 15:20
  • Found it. Sorry. I do mean to think more about this. – Betty Mock Nov 01 '13 at 20:40