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Re: Maclaurin series for ArcCosh[x]
*To*: mathgroup at smc.vnet.net
*Subject*: [mg73613] Re: Maclaurin series for ArcCosh[x]
*From*: "dimitris" <dimmechan at yahoo.com>
*Date*: Fri, 23 Feb 2007 04:31:07 -0500 (EST)
*References*: <erjoou$p9a$1@smc.vnet.net>
There is more mystery than originally I thought about!
Dimitris
Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
*This message was transferred with a trial version of CommuniGate(tm)
Pro*
Note that the first example can also be input as:
Assuming[x =C3=A2=CB=86=CB=86 Reals, ArcCosh[x] + O[x]^12]
SeriesData[x, 0, {(I/2)*Pi, -I, 0, -I/6, 0, (-3*I)/40, 0, (-5*I)/112,
0, (-35*I)/1152, 0, (-63*I)/2816}, 0, 12, 1]
This is consistent with ArcCosh[x] + O[x]^12 being equivalent to
Series[ArcCosh[x], {x, 0, 11}], with assumptions or without. So the
issue is why the answers are not the same when no assumptions about x
being real are present.
As for the second example:
since semantically TrigToExp[ArcCosh[x]] is exactly equivalent to
ArcCosh[x] it is natural to expect that
TrigToExp[ArcCosh[x]] + O[x]^12
and
ArcCosh[x]+ O[x]^12
return the same answer, but they do not. Which seems to me to bring
us back to my original "question", in a different form.
What this does seem to show, however, is that syntax and not just
semantics seems to matter here. In other words,
ArcCosh[x]+ O[x]^12
is not evaluated by simply automatically converting it to
Series[ArcCosh[x], {x, 0, 11}]
as I have always (until now) believed.
Note that in the case of an undefined function f the two forms of
input always evaluate to the same thing, e.g.
FullForm[f[x] + O[x]^2]
FullForm[SeriesData[x, 0, {f[0], Derivative[1][f][0]}, 0, 2, 1]]
FullForm[Series[f[x], {x, 0, 1}]]
FullForm[SeriesData[x, 0, {f[0], Derivative[1][f][0]}, 0, 2, 1]]
Clearly this is not what happens when f is ArcCosh, otherwise the two
answers would be the same. So I assume that in this case for the input
ArcCosh[x]+ O[x]^12
some sort of "parsing" fails and the expression is not converted into
one of the form Series[ ]. However, since no message is issued but
simply the original input is returned back (which I don't think
should ever happen in such cases) I suspect that something unintended
has occured.
Andrzej Kozlowski
On 22 Feb 2007, at 13:32, dimitris anagnostou wrote:
> Ok I understand.
>
> So, what will comment on the following?
>
> Series[ArcCosh[x], {x, 0, 11}, Assumptions -> Element[x,Reals]]
> SeriesData[x, 0, {(I/2)*Pi, -I, 0, -I/6, 0, (-3*I)/40, 0, (-5*I)/112,
> 0, (-35*I)/1152, 0, (-63*I)/2816}, 0, 12, 1]
>
> TrigToExp[ArcCosh[x]] + O[x]^12
> SeriesData[x, 0, {(I/2)*Pi, -I, 0, -I/6, 0, (-3*I)/40, 0, (-5*I)/112,
> 0, (-35*I)/1152, 0, (-63*I)/2816}, 0, 12, 1]
>
>
> Andrzej Kozlowski wrote:
> *This message was transferred with a trial version of CommuniGate
> (tm) Pro*
>
> On 22 Feb 2007, at 11:35, dimitris anagnostou wrote:
>
> >
> > In[47]:=
> > $Version
> >
> > Out[47]=
> > "5.2 for Microsoft Windows (June 20, 2005)"
> >
> > I think you have encountered something I would say is not a bug but
> > rather a feature.
>
>
> If it is a "feature" it is certainly a very well hidden one. However,
> I can see no justification for this in your post; and, to me it
> seesms clear that the two forms of input, by means of Series and + O
> [x]^n, have always been meat be equivalent, see for exmample seciton
> 3.6.2 of the Mathematica book, and particularly the sentence:
>
> Any time that an object like O[x] appears in a sum of terms,
> Mathematica will in fact convert the whole sum into a power series.
>
> "Features" should be made of sterner stuff.
>
> Andrzej Kozlowski
>
>
> >
> > Anyway, I believe that, the problematic behavior is due to the
> > presence of Floor function in the series expansion
> > not only of ArcCosh but also of ArcCot[x], ArcCoth[x], ArcCsc[x],
> > ArcCsch[x], ArcSec[x] nad ArcSech[x].
> >
> > The following commands will demontrate that
> >
> > In[48]:=
> > ToExpression[Names["Arc*"]]
> > Through[%[x]]
> > ({#1, Series[#1, {x, 0, 11}]} & ) /@ %
> > ({#1, #1 + O[x]^12} & ) /@ %%
> >
> > Out[48]=
> > {ArcCos, ArcCosh, ArcCot, ArcCoth, ArcCsc, ArcCsch, ArcSec, ArcSech,
> > ArcSin, ArcSinh, ArcTan, ArcTanh}
> >
> > Out[49]=
> > {ArcCos[x], ArcCosh[x], ArcCot[x], ArcCoth[x], ArcCsc[x], ArcCsch
> [x],
> > ArcSec[x], ArcSech[x], ArcSin[x], ArcSinh[x], ArcTan[x], ArcTanh[x]}
> >
> > Out[50]=
> > {{ArcCos[x], SeriesData[x, 0, {Pi/2, -1, 0, -1/6, 0, -3/40, 0,
> -5/112,
> > 0, -35/1152, 0, -63/2816}, 0, 12, 1]},
> > {ArcCosh[x], (-1)^Floor[Arg[x]/(2*Pi)]*SeriesData[x, 0, {(I/2)*Pi, -
> > I, 0, -I/6, 0, (-3*I)/40, 0, (-5*I)/112, 0, (-35*I)/1152,
> > 0, (-63*I)/2816}, 0, 12, 1]}, {ArcCot[x], (1/2)*(-1)^Floor[(Pi +
> > 2*Arg[x])/(2*Pi)]*Pi +
> > SeriesData[x, 0, {-1, 0, 1/3, 0, -1/5, 0, 1/7, 0, -1/9, 0, 1/11},
> > 1, 12, 1]},
> > {ArcCoth[x], (-I)*((1/2)*(-1)^Floor[Arg[x]/Pi]*Pi + SeriesData[x, 0,
> > {I, 0, I/3, 0, I/5, 0, I/7, 0, I/9, 0, I/11}, 1, 12,
> > 1])}, {ArcCsc[x], (1/2)*I*(-1)^Floor[Arg[x]/
> > Pi]*(-2*I*Pi*Floor[Arg[x]/Pi] +
> > SeriesData[x, 0, {(-I)*Pi - Log[4] + 2*Log[x], 0, 1/2, 0, 3/16,
> > 0, 5/48, 0, 35/512, 0, 63/1280}, 0, 12, 1])},
> > {ArcCsch[x], (-(1/2))*(-1)^Floor[(Pi + 2*Arg[x])/
> > (2*Pi)]*(-2*I*Pi*Floor[(Pi + 2*Arg[x])/(2*Pi)] +
> > SeriesData[x, 0, {-Log[4] + 2*Log[x], 0, -1/2, 0, 3/16, 0, -5/48,
> > 0, 35/512, 0, -63/1280}, 0, 12, 1])},
> > {ArcSec[x], Pi/2 - (1/2)*I*(-1)^Floor[Arg[x]/
> > Pi]*(-2*I*Pi*Floor[Arg[x]/Pi] +
> > SeriesData[x, 0, {(-I)*Pi - Log[4] + 2*Log[x], 0, 1/2, 0, 3/16,
> > 0, 5/48, 0, 35/512, 0, 63/1280}, 0, 12, 1])},
> > {ArcSech[x], (-(1/2))*I*(-1)^Floor[Arg[x]/Pi]*Pi +
> > (1/2)*(2*I*Pi*Floor[Arg[x]/Pi] + SeriesData[x, 0, {I*Pi + Log[4] -
> > 2*Log[x]}, 0, 12, 1]) +
> > SeriesData[x, 0, {-1/4, 0, -3/32, 0, -5/96, 0, -35/1024, 0,
> > -63/2560}, 2, 12, 1]},
> > {ArcSin[x], SeriesData[x, 0, {1, 0, 1/6, 0, 3/40, 0, 5/112, 0,
> > 35/1152, 0, 63/2816}, 1, 12, 1]},
> > {ArcSinh[x], SeriesData[x, 0, {1, 0, -1/6, 0, 3/40, 0, -5/112, 0,
> > 35/1152, 0, -63/2816}, 1, 12, 1]},
> > {ArcTan[x], SeriesData[x, 0, {1, 0, -1/3, 0, 1/5, 0, -1/7, 0, 1/9,
> > 0, -1/11}, 1, 12, 1]},
> > {ArcTanh[x], SeriesData[x, 0, {1, 0, 1/3, 0, 1/5, 0, 1/7, 0, 1/9, 0,
> > 1/11}, 1, 12, 1]}}
> >
> > Out[51]=
> > {{ArcCos[x], SeriesData[x, 0, {Pi/2, -1, 0, -1/6, 0, -3/40, 0,
> -5/112,
> > 0, -35/1152, 0, -63/2816}, 0, 12, 1]},
> > {ArcCosh[x], ArcCosh[x] + SeriesData[x, 0, {}, 12, 12, 1]},
> > {ArcCot[x], ArcCot[x] + SeriesData[x, 0, {}, 12, 12, 1]},
> > {ArcCoth[x], ArcCoth[x] + SeriesData[x, 0, {}, 12, 12, 1]},
> > {ArcCsc[x], ArcCsc[x] + SeriesData[x, 0, {}, 12, 12, 1]},
> > {ArcCsch[x], ArcCsch[x] + SeriesData[x, 0, {}, 12, 12, 1]},
> > {ArcSec[x], ArcSec[x] + SeriesData[x, 0, {}, 12, 12, 1]},
> > {ArcSech[x], ArcSech[x] + SeriesData[x, 0, {}, 12, 12, 1]},
> > {ArcSin[x], SeriesData[x, 0, {1, 0, 1/6, 0, 3/40, 0, 5/112, 0,
> > 35/1152, 0, 63/2816}, 1, 12, 1]},
> > {ArcSinh[x], SeriesData[x, 0, {1, 0, -1/6, 0, 3/40, 0, -5/112, 0,
> > 35/1152, 0, -63/2816}, 1, 12, 1]},
> > {ArcTan[x], SeriesData[x, 0, {1, 0, -1/3, 0, 1/5, 0, -1/7, 0, 1/9,
> > 0, -1/11}, 1, 12, 1]},
> > {ArcTanh[x], SeriesData[x, 0, {1, 0, 1/3, 0, 1/5, 0, 1/7, 0, 1/9, 0,
> > 1/11}, 1, 12, 1]}}
> >
> >
> > Best Regards
> > Dimitris
=CE=9F/=CE=97 Andrzej Kozlowski =CE=AD=CE=B3=CF=81=CE=B1=CF=88=CE=B5:
> Try:
>
> Series[ArcCosh[x], {x, 0, 11}]
>
> and now try
>
> ArcCosh[x] + O[x]^12
>
> At least with my version of Mathematica:
>
> $Version
> 5.2 for Mac OS X (February 24, 2006)
>
>
> I do not get the same answer (in fact in the latter case the input is
> returned unevaluated). With ArcSinh and any other function that I
> have tried in place of ArcCosh the outputs are always the same.
>
> Andrzej Kozlowski
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