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Re: [Axiom-developer] One more integral to add to the tests, if not alre
From: |
Martin Rubey |
Subject: |
Re: [Axiom-developer] One more integral to add to the tests, if not already there |
Date: |
18 Oct 2006 08:15:27 +0200 |
User-agent: |
Gnus/5.09 (Gnus v5.9.0) Emacs/21.3 |
"William Sit" <address@hidden> writes:
> On Tue, 17 Oct 2006 17:18:37 -0700 (PDT)
> C Y <address@hidden> wrote:
> >integrate(%e^cos(x)*cos(x-sin(x)),x = 0..2*%pi)
> >
> > Apparently this can be solved by Mathematica in it's upcoming version. We
> > currently get "failed."
> >
> >Cheers,
> >CY
>
> Apparently, Mathematica got this wrong in Version 5.2:
> In[1]:=
> Integrate[Exp[Cos[x]] Cos[x-Sin[x]], {x,0,2 Pi}]
>
> Out[1]=
> 0
>
> But a look at the graph of the integrand shows this is clearly wrong. Indeed,
> In[3]:=
> NIntegrate[Exp[Cos[x]] Cos[x-Sin[x]], {x,0,2 Pi}]
>
> Out[3]=
> 6.28319
>
> which looks like the exact answer is 2 Pi.
Yes, both Maple and Mathematica currently need help (namely you have to say
Expand[Cos[x - Sin[x]]]), but the upcoming version of Mathematica fixes this.
See:
http://groups.google.com/group/sci.math.symbolic/browse_frm/thread/2987d75001d88189/408eefdbb9b6266b?lnk=raot&hl=en#408eefdbb9b6266b
It would really be good to have an integration expert around.
Martin
By the way, the indefinite integral is according the MMA
In[2]:= Integrate[E^Cos[x] * TrigExpand[ Cos[ x - Sin[x] ] ], x]
I x -I x
I E - I x E + I x -I x
Out[2]= - (E - E + ExpIntegralEi[E ] -
2
I x
> ExpIntegralEi[E ])
so I suspect that we might be able to tweak the pattern matcher to do it.
TrigExpand gives
In[3]:= TrigExpand[ Cos[ x - Sin[x] ] ]
Cos[x] I Cos[x] I
Out[3]= Cos[x] Cosh[------ - - Sin[x]] Cosh[------ + - Sin[x]] +
2 2 2 2
Cos[x] I Cos[x] I
> I Cosh[------ + - Sin[x]] Sin[x] Sinh[------ - - Sin[x]] -
2 2 2 2
Cos[x] I Cos[x] I
> I Cosh[------ - - Sin[x]] Sin[x] Sinh[------ + - Sin[x]] -
2 2 2 2
Cos[x] I Cos[x] I
> Cos[x] Sinh[------ - - Sin[x]] Sinh[------ + - Sin[x]]
2 2 2 2
Maybe you could look into this, Cliff?
You will have to consider the proposed fix to issue #191
Martin