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Re: Re: asymptotics
This gives me an opportunity to write a few more words on asymptotic
expansions, including soem things that I used to know but forgot.
There are in fact several concepts of defintions of "asymptotic
expansions". The most common defintion is that of an asymptotic-power
series, of the form
f(x) = a[0] + a[1] x^(-1) + ... + a[n-1] x^(-n+1) + (a[n]+epsion[n]
[x]) x^-n
where epsilon[n][x]->0 as x->0.
This says that the difference between the sum of the first n-terms
of the expansion and the function f[x] can be made as small as
possible in comparison with 1/x^n when x is large enough.
In the case of this kind of expansions we have a certain degree of
uniqueness: a given functions can have only one expansion for a given
range of values of x, but two different functions can have the same
asymptotic expansion. For example, the asymptotic expansion of
Exp[-x]
is
0+0+..... so the asymptotic expansion of f[x] and f[x]+ Exp[-x] will
be the same (however, Mathematica, rather naturally will not give you
any expansion of Exp[-x] at Infinity - if it did it would have to
return 0).
On the other hand, Exp[x], has not asymptotic expansion at all.
Another common kind (about which I forgot but Maxim Rytin remembered)
is the case f[x]/g[x]. If f[x]/g[x] has an expansion of the above
kind then one says that
f(x) = g[x] (a[0] + a[1] x + ... + a[n-1]/x^(n-1) + ...)
is an asymptotic expansion of f. This is the kind of expansion that
the other CAS gave you and it has, in general, better properties than
the ones I considered.
There are, however, much more general asymptotic expansions, which
is what i had in mind in my first replies to you. In fact, one can
expand functions in terms of essentially any given functions. I
think the most general "theory" of this kind was invented by Hoene-
Wronski but it has never been given a formal mathematical basis. The
subject is very complicated and obscure with many special techniques
and only very limited general theory. It does not seem to be suitable
for CAS.
Now finally for your question. To tell the truth I do not know the
answer. When asked to expand a Bessel function at Infinity
Mathematica first produces messages about essential singularities
and then returns an expansion that looks like one of the expansions
you can see here
http://functions.wolfram.com/BesselAiryStruveFunctions/BesselJ/06/02/02/
Probably the explanation is that in the case of well known special
functions like the Bessel functions someone has programmed
Mathematica to return these expansions while it makes no attempt to
find them in more general cases. But actually, it seems, the all that
the other CAS does is simply to also consider the case:
g[x]*(something that has a power series at infinity), where g[x] has
an essential singularity at Infinity. This kind of thing is simple
enough to do by oneself.
Andrzej Kozlowski
On 28 May 2007, at 13:59, dimitris wrote:
> Hello.
>
> Hmmm!
>
>> 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].
>
> Because of applied mathematics background I may miss something
> fundamentally.
> So I apologize for it.
>
> Why for example Mathematica does expand BesselJ at infinity?
> There is an essential singularity there.
>
> Dimitris
>
> / Andrzej Kozlowski :
>> 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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