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)