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Re: simple question

• To: mathgroup at smc.vnet.net
• Subject: [mg73705] Re: simple question
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Mon, 26 Feb 2007 06:07:16 -0500 (EST)
• References: <20070225101155.493$AH_-_@newsreader.com>

On 25 Feb 2007, at 16:11, David W. Cantrell wrote:

> [Message also posted to: comp.soft-sys.math.mathematica]
>
> Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
>> Sorry, I posted not what I had intended. Let me do it all again. We
>> define:
>>
>> I1 = Pi/2 - ArcTan[u];
>> I2 = ArcTan[1/u];
>>
>> and want to show that I1==I2 for u>0. We check that the derivative of
>> the difference is 0:
>>
>> In[42]:=
>> FullSimplify[D[I1-I2,u],u>0]
>>
>> Out[42]=
>> 0
>>
>> and not, to prove that I1==I2 for u>0 we simply need to evaluate the
>> difference I1-I2 for just one suitable chosen u>0, e.g.
>>
>> FullSimplify[(I1-I2)/.u->1]
>>
>> 0
>
> But one must be careful with that technique! Note that _everything_  
> above
> still seems to work perfectly if the restriction u>0 is simply omitted
> throughout, thereby seeming to show that Pi/2 - ArcTan[u] == ArcTan 
> [1/u]
> for _all_ u, which is, of course, false.
>
> David

Yes, indeed.  A combination of Mathematica and mathematics can be  
very powerful. Alone each has its limitations, but using Mathematica  
without paying attention to mathematics can be dangerous.

Andrzej Kozlowski