       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:=
Integrate[Abs[x], x]

Out=
Integrate[Abs[x], x]

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

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

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

Out=
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:= Integrate[Abs[Cos[x]],x]
>
> Out= Integrate[Abs[Cos[x]], x]
>
> In:= Integrate[Abs[Cos[u]],{u,0,x}]
>
> 2
> Out= Sqrt[Cos[x] ] Tan[x]
>
> In:= 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:= Integrate[Abs[Cos[x]],{x,0,y}]
>
> 2
> Out= Sqrt[Cos[y] ] Tan[y]
>
> In:= Integrate[Abs[Cos[x]],{x,0,y}] // InputForm
>
> Out//InputForm= Sqrt[Cos[y]^2]*Tan[y]
>
> In:= D[Sqrt[Cos[y]^2]*Tan[y],y] // Simplify
>
> 2
> Out= Sqrt[Cos[y] ]
>
> In:= Quit
>
>
>> 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= 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.
>>
>>
>> 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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