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Re: Power Series for LogIntegral

  • To: mathgroup at smc.vnet.net
  • Subject: [mg45225] Re: Power Series for LogIntegral
  • From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
  • Date: Sun, 21 Dec 2003 03:42:17 -0500 (EST)
  • References: <bs1bgg$9n3$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

drbob at bigfoot.com (Bobby R. Treat) wrote:
> Why doesn't Series return a power series?
>
> Series[LogIntegral[x], {x, 0, 1}]
>
> SeriesData[x, 0,
> {6/Log[x]^4 + 2/Log[x]^3 + Log[x]^(-2) + Log[x]^(-1)}, 1, 2, 1]

The trivial answer is that Mathematica can't return what doesn't exist:
LogIntegral[x] does not have a Maclaurin series.

Mathematica does, however, return a portion of a Puiseux series, which
could potentially be useful to you.

But what I find curious is that Mathematica chose to give terms up to power
_4_ of Log[x] in the denominator. Why go up to _that_ power? And why stop
_there_? It makes no sense to me. If I had had to guess what Mathematica
would give, I would probably have guessed that it would give just

SeriesData[x, 0, {Log[x]^(-1)}, 1, 2, 1]

BTW, I raised similar questions earlier this month in the thread "Mistake
about ProductLog expansion at Infinity".

David Cantrell


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