Mathematica - Maclaurin series for ArcCosh[x]

Author

Maclaurin series for ArcCosh[x]

Andrzej Kozlowski

2007-02-22, 4:24 am

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

Jean-Marc Gulliet

2007-02-23, 4:23 am

Andrzej Kozlowski wrote:
> 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.

Hi Andrzej,

Same behavior on my system.

In[1]:=
$Version

Out[1]=
5.2 for Microsoft Windows (June 20, 2005)

In[2]:=
Series[ArcCosh[x], {x, 0, 11}]

Out[2]=
Floor[Arg[x]/(2 Pi)]
(-1)

SeriesData[Global`x, 0,

I -I -3 I -5 I -35 I
{- Pi, -I, 0, --, 0, ----, 0, ----, 0, -----, 0,
2 6 40 112 1152

-63 I
-----}, 0, 12, 1]
2816

In[3]:=
ArcCosh[x] + O[x]^12

Out[3]=
ArcCosh[x] + SeriesData[Global`x, 0, {}, 12, 12, 1]

In[4]:=
Series[ArcSinh[x], {x, 0, 11}]

Out[4]=
SeriesData[Global`x, 0,

1 3 5 35
{1, 0, -(-), 0, --, 0, -(---), 0, ----, 0,
6 40 112 1152

63
-(----)}, 1, 12, 1]
2816

In[5]:=
ArcSinh[x] + O[x]^12

Out[5]=
SeriesData[Global`x, 0,

1 3 5 35
{1, 0, -(-), 0, --, 0, -(---), 0, ----, 0,
6 40 112 1152

63
-(----)}, 1, 12, 1]
2816

Regards,
Jean-Marc

dimitris

2007-02-23, 4:23 am

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@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

dimitris

2007-02-23, 4:23 am

There is more mystery than originally I thought about!

Dimitris

Andrzej Kozlowski <akoz@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:
>
>
>
> 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
>
>
> [x],
> -5/112,
> -5/112,

=CE=9F/=CE=97 Andrzej Kozlowski =CE=AD=CE=B3=CF=81=CE=B1=CF=88=CE=B5:[co
lor=darkred]
> 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

David W.Cantrell

2007-02-23, 4:23 am

Andrzej Kozlowski <akoz@mimuw.edu.pl> wrote:
> 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.

FWIW:
I think this does not happen because of ArcCosh in particular. For example,
Series[ArcCosh[x], {x, 1, 11}] and ArcCosh[x] + O[x - 1]^12 _do_ give
the same answer. I must guess the quirk you observed is due to the nature
of the series. Based on that guess, I next tried
Series[ArcCot[x], {x, 0, 11}] and ArcCot[x] + O[x]^12 and so immediately
found another example of that quirk.

And, Andrzej, thank you for your suggestion, in an earlier thread, about
showing that a certain hypergeometric expression is 0.

David

dimitris

2007-02-23, 4:23 am

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.

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

> 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

Bill Rowe

2007-02-23, 4:23 am

On 2/22/07 at 4:35 AM, akoz@mimuw.edu.pl (Andrzej Kozlowski) wrote:

>Try:

>Series[ArcCosh[x], {x, 0, 11}]

I get series expansion as expected

>and now try

>ArcCosh[x] + O[x]^12

It isn't obvious to me what you are suggesting here. If I enter

ArcCosh[x] + O[x]^12

into a new cell, I get that returned as I would expect

If I enter

Series[ArcCosh[x] + O[x]^12, {x, 0, 11}]

into a new cell I get the same result as for

Series[ArcCosh[x],{x, 0, 11}]

except there is now an additinonal O[x]^12 term in front of the
previous result. This seems reasonbable.

>At least with my version of Mathematica:

>$Version 5.2 for Mac OS X (February 24, 2006)

I am using a slightly different version

$Version 5.2 for Mac OS X (June 20, 2005)

but it isn't clear this is significant
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