Re: Re: Strange results
- To: mathgroup at smc.vnet.net
- Subject: [mg15436] Re: [mg15421] Re: [mg15389] Strange results
- From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
- Date: Mon, 18 Jan 1999 04:21:41 -0500
- Sender: owner-wri-mathgroup at wolfram.com
Just a small correction to my reply. In it I unfairly blamed Mathemtica
for giving incorrect answer for the case n=0 and certain ranges of x
but actually Mathematica gives the correct answer:
n
p = Integrate[Cos[x] , x];
In[3]:=
Simplify[p /. n -> 0]
Out[3]=
2 2
ArcSin[Sqrt[Cos[x] ]] Sqrt[Cos[x] ] Tan[x]
-(------------------------------------------)
2
Sqrt[Sin[x] ]
Now,for example if 0<=x<=Pi (Sin[x]>0) this becomes -ArcSin[Cos[x]]
while for Pi<=x<=2Pi (Sin[x]<0) this gives ArcSin[Cos[x]]. In either
case the answer is correct. The problem in this case was entirely due
to the use of PowerExpand.
On Wed, Jan 13, 1999, Andrzej Kozlowski <andrzej at tuins.ac.jp> wrote:
>The answer Mathematica gives you for n=1 (-1+Sin[x]) is completely
>equivalent to what you expected (Sin[x]) since they differ by a
>constant and thus have the same derivative: Cos[x]. The situation is
>somewhat more complicated in the case of n=0. The function
>-ArcSin[Cos[x]] =x-Pi/2 over certain ranges (e.g. 0=<x=<Pi), over
>others it is Pi/2-x (e.g. -Pi=<x<=0).
>
>It is easy to see why you get this strange answer. Mathematica tries to
>give you the most general answer it can find. It can't give you a
>single aswer that works for all possible values of n, and x so it gives
>you a "generic" one. The answer it gives is not valid for n=-1 for n=0
>and certain ranges of x, but other than that it is fine. You may not be
>able to see this at once, but you can always check it by taking
>derivatives. For example, for n= -1/2 you get
>
>In[1]:=
> 1
>v = p /. n -> -(-)
> 2
>Out[1]=
> 1 1 5 2
> 2 Sqrt[Cos[x]] Hypergeometric2F1[-, -, -, Cos[x] ] Sin[x]
> 4 2 4
>-(---------------------------------------------------------)
> 2
> Sqrt[Sin[x] ]
>
>
>While on the other hand:
>
>In[2]:=
> -(1/2)
>w = p = Integrate[Cos[x] , x]
>Out[2]=
> x
>2 EllipticF[-, 2]
> 2
>
>
>These answers look different but
>
>
>In[3]:=
>Simplify[D[v - w, x]]
>Out[3]=
>0
>
>On Tue, Jan 12, 1999, Ing. Alessandro Toscano Dr. <toscano at ieee.org>
>wrote:
>
>>The following in/out does not make sense to me:
>>
>>In[2]:=
>>p=Integrate[Cos[x]^n,x]
>>Out[2]=
>>\!\(\(-\(\(Cos[x]\^\(1 + n\)\
>> Hypergeometric2F1[\(1 + n\)\/2, 1\/2, \(3 + n\)\/2,
>>Cos[x]\^2]\
>> Sin[x]\)\/\(\((1 + n)\)\ \ at Sin[x]\^2\)\)\)\) In[4]:=
>>p//.n->0//PowerExpand
>>Out[4]=
>>-ArcSin[Cos[x]]
>>
>>In[7]:=
>>p//.n->1//Simplify//PowerExpand
>>Out[7]=
>>-1+Sin[x]
>>
>>
>>Isn't it true that (Integrate[Cos[x]^0,x] == x? Isn't it true that
>>(Integrate[Cos[x]^1,x] == Sin[x]?
>>
>>
>>Why do I get this strange result?
>>
>>I am using Mathematica 3.01 on Pcs.
>>
>>Thanks for any info.
>>
>>
>>***********************************
>>Ing. Alessandro Toscano Dr.
>>
>>Universite di Roma Tre
>>Dip. Ingegneria Elettronica
>>Via della Vasca Navale, 84
>>00146, Roma, ITALIA
>>
>>Tel. +39-6-55177095
>>Fax +39-6-5579078
>>mailto:toscano at ieee.org
>><http://ato.ele.uniroma3.it>
>>
>>************************************
>
>
>Andrzej Kozlowski
>Toyama International University
>JAPAN
>http://sigma.tuins.ac.jp/
>http://eri2.tuins.ac.jp/
>
Andrzej Kozlowski
Toyama International University
JAPAN
http://sigma.tuins.ac.jp/
http://eri2.tuins.ac.jp/