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Re: SeriesCoefficient: needs work!
*To*: mathgroup at smc.vnet.net
*Subject*: [mg83490] Re: SeriesCoefficient: needs work!
*From*: Peter Pein <petsie at dordos.net>
*Date*: Wed, 21 Nov 2007 03:01:57 -0500 (EST)
*References*: <fhrr3r$53r$1@smc.vnet.net>
jackgold at umich.edu schrieb:
> Hi Everyone,
>
> I have been experimenting with the new (ver 6) command,
> SeriesCoefficient in the form,
>
> SeriesCoefficient[fnt,{x,x0,n}].
>
> Here fnt is a function of x and n is symbolic. This command is
> supposed to return the nth coefficient in the series expansion of fnt
> about x0.
>
> I have found the following results on a MacBook Pro running Tiger.
>
> 1) SeriesCoefficient[Cos[x] Exp[x], {x, 0, n}] returns itself, unevaluated.
>
> 2) SeriesCoefficient[Cos[x] Exp[x]/(1-x), {x, 0, n}] returns an
> expression involving incomplete Gamma functions with an imaginary
> argument. Odd that 1) does not compute but 2) does! Not that the
> result is terribly revealing, by the way.
>
> 3) SeriesCoefficient[Sin[x] Exp[x], {x, 0, n}] so preposterously
> complicated that most of us would have preferred no result! (Just
> joking. The result is far to complicated to publish here.)
>
> My opinion is that this use of SeriesCoefficient should not be offered
> to the public until some of these obvious glitches are cleaned up.
> Incidentally, since the nth terms of the individual functions Sin[x],
> Cos[x], Exp[x] and 1/(1-x) can be found using SeriesCoefficient and
> surely Mathematica knows how to find the nth coefficient of a product of power
> series, I suspect the problem lies in the finite summation which
> results from the use of the Cauchy product formula.
>
>
Strange.
In Version 5.2 the good old SeriesTerm[] in the package DiscreteMath got its
difficulties in case 2:
In[1]:=
<< "DiscreteMath`"
In[2]:=
SeriesTerm[Cos[x]*E^x, {x, 0, n}]
Out[2]=
(1 - I)^n/(2*n!) + (1 + I)^n/(2*n!)
In[3]:=
SeriesTerm[(Cos[x]*E^x)/(1 - x),
{x, 0, n}]
Out[3]=
Piecewise[{{ComplexInfinity, n >= 1}},
(I^n*KroneckerDelta[Mod[n, 2]])/
Gamma[1 + n]]
In[4]:=
SeriesTerm[Sin[x]*E^x, {x, 0, n}]
Out[4]=
-((1/Pi)*(2^(1 + n/2)*
(Cos[(n*Pi)/4] + Cos[(3*n*Pi)/4])*
Gamma[-n]*Sin[(n*Pi)/4]^2))
And the package SpecialFunctions by Prof. Wolfram Koepf (
http://www.mathematik.uni-kassel.de/~koepf/CA/index.html ) has got problems
with case 2 too:
In[1]:=
<< "SpecialFunctions`"
>From In[1]:=
"SpecialFunctions, (C) Wolfram Koepf, version 2.01, 2006"
>From In[1]:=
"Fast Zeilberger, (C) Peter Paule and Markus Schorn (V 2.2) loaded"
In[3]:=
PS[Cos[x]*Exp[x], x, 0]
Out[3]=
sum[(2^(k/2)*x^k*Cos[(k*Pi)/4])/k!, {k, 0, Infinity}]
In[4]:=
PS[Cos[x]*(Exp[x]/(1 - x)), x, 0]
Out[4]=
SpecialFunctions`Private`df[3]
In[5]:=
PS[Sin[x]*Exp[x], x, 0]
Out[5]=
sum[(2^(k/2)*x^k*Sin[(k*Pi)/4])/k!, {k, 0, Infinity}]
Shouldn't the V6-function SeriesCoefficient be at least as smart as the
V5.2-package is? But maybe the new version gets some series representations
where the old one fails.
Peter
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