       Re: asymptotics

• To: mathgroup at smc.vnet.net
• Subject: [mg76654] Re: asymptotics
• From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
• Date: Fri, 25 May 2007 06:36:10 -0400 (EDT)
• Organization: Uni Leipzig
• References: <f33qsc\$mc6\$1@smc.vnet.net>

```Hi,

the function has an essential singularity at u->Infinity and
remember the definition of an essential singularity
"A point a is an essential singularity if and only if the
limit Lim[f,z->a] does not exist as a complex number nor equals infinity"

and this will make it impossible to get the zero order
term of the series expansion.

An asymtotic power series and x->Infinity exist only if
for y[x] - Sum[a[n]*x^(-alpha*n),{n,0,N}] << x^(-alpha*N)
for x->Infinity and every N. And the exponential function
that you use has no powerseries expansion of this form.

So the CAS may give you a result but this is nonsense.

Regards
Jens

dimitris wrote:
> Sorry fellas if I ask something trivial
> but currently I can't find anything!
>
> In another CAS I took
>
> f:=asympt(exp(-y*sqrt(1+m^2*u^2)/m),u,5);
>
>        /        2 1/2      2          2 1/2      2    2
>        |    y (m )        y       y (m )    (-6 m  + y )
>   f := |1 - --------- + ------- - ----------------------
>        |        3          4  2              7  3
>        \     2 m  u     8 m  u           48 m  u
>
>             2       2    2           \             2 1/2
>            y  (-24 m  + y )      1   |   /     y (m )    u
>          + ---------------- + O(----)|  /  exp(-----------)
>                    8  4           5  | /            m
>               384 m  u           u   /
>
> ff:=simplify(convert(f,polynom)) assuming m>0;
> ff := 1/384*exp(-
> y*u)*(384*m^8*u^4-192*y*m^6*u^3+48*y^2*m^4*u^2+48*y*m^4*u-8*y^3*m^2*u-24*y^2*m^2+y^4)/
> m^8/u^4
>
> In Mathematica I can't get the expansion in infinity
>
> In:= Series[Exp[(-y)*(Sqrt[1 + m^2*u^2]/m)], {u, Infinity, 10}]
> Out= E^(-((Sqrt[1 + m^2*u^2]*y)/m))
>
> What do I miss here?
>
> Thanks
> Dimitris
>
>

```

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