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Re: asymptotics
*To*: mathgroup at smc.vnet.net
*Subject*: [mg76629] Re: [mg76613] asymptotics
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Fri, 25 May 2007 06:23:10 -0400 (EDT)
*References*: <200705241023.GAA21917@smc.vnet.net>
On 24 May 2007, at 19:23, 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
>
>
Well, of course, Mathematica correctly does not expand your function
as a power series about infinity because such an expansion does not
exist (or, if you prefer, is identically 0). Note that the
"assymtotic expansion" the other CAS gives you is into a power series
expansion: since after truncation it contains a factor Exp[-y u].
There are many such assymtotic expansions. I do not know how to use
Mathematica to get this particular one, but it is easy to get similar
ones. For example, here is one way to get an assymptotic expansion
pretty close to the one given by the other CAS:
f[u_] = Simplify[
Normal[Series[a^(Sqrt[m^2 + 1/u^2]/m), {u, Infinity, 10}]] /.
a -> Exp[(-y)*u], {m > 0, u > 0, y > 0}]
(3840*u^9*m^10 - 1920*u^8*y*m^8 + 480*u^6*y*(u*y + 1)*m^6 -
80*u^4*y*(u^2*y^2 + 3*u*y + 3)*m^4 +
10*u^2*y*(u^3*y^3 + 6*u^2*y^2 + 15*u*y + 15)*m^2 -
y*(u^4*y^4 + 10*u^3*y^3 + 45*u^2*y^2 + 105*u*y + 105))/(E^(u*y)
*(3840*
m^10*u^9))
This looks a little more complicated than the expression given by the
other CAS (and is, of course, not equal to it) but they both give
good approximations of the original function at Infinity. You can
check it numerically as follows:
g[u_] = 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;
Let's choose some random values for m and y:
m = Random[]; y = Random[];
then
f[20000] // N
5.318723018905205*10^-2317
g[20000] // N
5.318723018905206*10^-2317
which are pretty close. For larger values they are even closer:
f[200000] // N
1.811911734460420*10^-23163
g[200000] // N
1.811911734460420*10^-23163
Of course, as expected, the values are very close to zero. For the
function itself we get:
N[Exp[(-y)*(Sqrt[1 + m^2*u^2]/m)] /. u -> 200000]
1.81191173447088963591082`11.14761897993578*^-23163
Andrzej Kozlowski
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