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>
- Reply-to: kuska at informatik.uni-leipzig.de
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[113]:= Series[Exp[(-y)*(Sqrt[1 + m^2*u^2]/m)], {u, Infinity, 10}]
> Out[113]= E^(-((Sqrt[1 + m^2*u^2]*y)/m))
>
> What do I miss here?
>
> Thanks
> Dimitris
>
>
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