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MathGroup Archive 1998

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Re: A hard Series problem.

  • To: mathgroup at smc.vnet.net
  • Subject: [mg13755] Re: A hard Series problem.
  • From: lawry at maths.ox.ac.uk (James Lawry)
  • Date: Mon, 24 Aug 1998 05:07:07 -0400
  • Organization: Oxford Centre for Industrial & Applied Mathematics
  • References: <6rdltk$dmb@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Ersek, Ted R <ErsekTR at navair.navy.mil> wrote:
>Using the Limit package I was able to find the first several terms in a
>Laurent series of  Log[(1-Erf[x]) x Sqrt[x]]. The terms I was able to
>find are shown below.
>
>Log[(1-Erf[x])x Sqrt[Pi]]=
>	-x^2+(1/2)x^(-2)-(5/8)x^(-4)+(37/24)x^(-6)+(353/64)x^(-8)- ....
>
>The above uses Mathematica's definition of Erf, which is:
>Erf[x]=2/Sqrt[Pi] Integrate[Exp[-t^2],{t,0,x}]
>
>I ran into a brick wall when I tried to find other terms.
>
>I couldn't get the expression above using Series, but I was able to find
>the terms one at a time by using Limit.
>Any other ideas on how to determine the terms of the series?
>
>Can anyone find some more terms of the series? A general expression for
>the nth term of the series would be fabulous.

Here are the next few:

 ... - 4081/160 x^(-10) + 55205/384 x^(-12) - 854197/896 x^(-14) +
14876033/2048 x^(-16) - 288018721/4608 x^(-18) + .....

I didn't use the Limit package, just the following command:

Log[Series[Erfc[x], {x, Infinity, 20}] Exp[x^2] x Sqrt[Pi]]

Mathematica complains about essential singularities but still computes
the series ok. Change 20 as appropriate for more terms. Sorry, I don't
have a general term expression. 

Note that this is _not_ a Laurent series, it's an asymptotic expansion
for large |x| and is only valid in either the sector -3 Pi/4 < Arg[x] <
3 Pi/4 (the anti-Stokes line) or the sector -Pi/2 < Arg[x] < Pi/2 (the
Stokes line), depending on your definition of "asymptotic". (That's why
Mathematica complains so vigorously about doing the manipulation.)

James Lawry.
(emailed and posted to newsgroup)


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