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