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MathGroup Archive 2007

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Re: Maclaurin series for ArcCosh[x]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg73602] Re: Maclaurin series for ArcCosh[x]
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Fri, 23 Feb 2007 04:25:09 -0500 (EST)
  • References: <erjoou$p9a$1@smc.vnet.net>

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 <akoz at mimuw.edu.pl> 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=2E6.2 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

=CF/=C7 Andrzej Kozlowski =DD=E3=F1=E1=F8=E5:
> 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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