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MathGroup Archive 2000

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Re: Re: Simple integral wrong

  • To: mathgroup at smc.vnet.net
  • Subject: [mg25160] Re: [mg25107] Re: Simple integral wrong
  • From: Andrzej Kozlowski <andrzej at bekkoame.ne.jp>
  • Date: Tue, 12 Sep 2000 02:58:47 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

This is of course very easy to explain and exactly as expected.
Integrate[Abs[Cos[u]],{u,0,x}] is a path integral of a non-analytic function
and not an indefinite integral("anti-derivative"). The fundamental theorem
of calculus does not hold for path integrals of non-analytic functions. This
should not be a surprise since it is only standard undergraduate
mathematics.  For an even more forceful demonstration look at this example:

In[2]:=
Integrate[Abs[x], x]

Out[2]=
Integrate[Abs[x], x]

In[3]:=
Integrate[Abs[x], {x, 0, z}]

Out[3]=
1             2        2
- z Sqrt[Im[z]  + Re[z] ]
2

In[4]:=
D[%, z] // FullSimplify

Out[4]=
     2        2
Im[z]  + Re[z]  + z Im[z] Im'[z] + z Re[z] Re'[z]
-------------------------------------------------
                    2 Abs[z]


By the way, there is nothing wrong with the path integral here, provided one
takes as the path the straight line from 0 to z. The differentiation is
meaningless, since Re and Im are not differentiable, but (in my opinion) you
can't blame Mathematica for giving a meaningless answer to a meaningless
question.

on 00.9.10 4:14 PM, Albert Retey at albert.retey at visualanalysis.com wrote:

> Hi all,
> 
> This leaves even more questions, actually I would prefer the first
> "result"...
> 
> Mathematica 4.0 for Linux
> Copyright 1988-1999 Wolfram Research, Inc.
> -- Motif graphics initialized --
> 
> In[1]:= Integrate[Abs[Cos[x]],x]
> 
> Out[1]= Integrate[Abs[Cos[x]], x]
> 
> In[2]:= Integrate[Abs[Cos[u]],{u,0,x}]
> 
> 2
> Out[2]= Sqrt[Cos[x] ] Tan[x]
> 
> In[3]:= Quit
> 
> 
> Note that also the Differentiation goes wrong (which might be th reason
> for the wrong Integration after all):
> 
> Mathematica 4.0 for Linux
> Copyright 1988-1999 Wolfram Research, Inc.
> -- Motif graphics initialized --
> 
> In[1]:= Integrate[Abs[Cos[x]],{x,0,y}]
> 
> 2
> Out[1]= Sqrt[Cos[y] ] Tan[y]
> 
> In[2]:= Integrate[Abs[Cos[x]],{x,0,y}] // InputForm
> 
> Out[2]//InputForm= Sqrt[Cos[y]^2]*Tan[y]
> 
> In[3]:= D[Sqrt[Cos[y]^2]*Tan[y],y] // Simplify
> 
> 2
> Out[3]= Sqrt[Cos[y] ]
> 
> In[4]:= Quit
> 
> 
>> I have had a number or queries about this. Sorry I hadn't made it clearer.
>> Here is
>> the full story:
>> 
>> A couple of people told me that
>> 
>> Plot[Integrate[Abs[Cos[u]], {u, 0, x  Pi]}], {x, 0, 3}]
>> 
>> works fine. The result is monotonic increasing as expected.
>> 
>> But try
>> 
>> Plot[Evaluate[Integrate[Abs[Cos[u]],{u,0,Pi*x}]],{x,0,3}]
>> 
>> and see what happens! The evaluate forces Mathematica to do the
>> integral symbolically. It was doing it numerically without the Evaluate.
>> 
>> Or just type
>> 
>> Integrate[Abs[Cos[u]],{u,0,Pi x}]
>> 
>> Mathematica 4 returns
>> 
>> 2
>> Out[1]= Sqrt[Cos[Pi x] ] Tan[Pi x]
>> 
>> (Actually, I don't think Mathematica 3 can do it at all.) This
>> plots as a saw-tooth. The true solution should be
>> 
>> Sqrt[Cos[Pi x]^2] Tan[Pi x] + 2 Floor[x + 1/2]
>> 
>> Mathematica misses the step functions necessary to make the solution
>> continuous.
>> 
>> Thanks for your interest,
>> 
>> Paul Cally
>> 
>> --
>> 
>> +--------------------------------------------------------------------------+
>> |Assoc Prof Paul Cally            |    Ph:  +61 3 9905-4471                |
>> |Dept of Mathematics & Statistics |    Fax: +61 3 9905-3867                |
>> |Monash University                |    paul.cally at sci.monash.edu.au        |
>> |PO Box 28M, Victoria 3800        |                                        |
>> |AUSTRALIA                        | http://www.maths.monash.edu.au/~cally/ |
>> +--------------------------------------------------------------------------+
> 
> --
> Visual Analysis GmbH     Internet: www.visualanalysis.com
> Neumarkter Str. 87       Telefon: 089 / 431 981 0
> D-81673 Muenchen         Telefax: 089 / 431 981 1
> 

-- 
Andrzej Kozlowski
Toyama International University
JAPAN

http://platon.c.u-tokyo.ac.jp/andrzej/
http://sigma.tuins.ac.jp/



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