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Re: The domain parameter of Reduce[]

• To: mathgroup at smc.vnet.net
• Subject: [mg124163] Re: The domain parameter of Reduce[]
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Wed, 11 Jan 2012 04:26:32 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201201101059.FAA27797@smc.vnet.net>
• Reply-to: murray at math.umass.edu

Manifestly, the result of the first Reduce, namely,

   a > 0 && b > 0

is correct whereas the result of the second Reduce, namely,

   a > 0 && 0 < b <= a

is wrong. For example,

   Re[-1 - Sqrt[a - b] /. {a -> 1/4, b -> 1/2}]

has value -1 even though b > a.

But I don't know why adding the domain Complexes works, because I've 
never understood the documentation for Reduce as to exactly what is mean 
by "doing the Reduction over" a specified domain.

The documentation seems to suggest by means of examples that what this 
means is that the equalities and inequalities are allowed to hold in 
that domain -- rather than that the variables involved are restricted 
to, or allowed to range over values, in that domain.

But I don't think the docs actually come right out and say that. (An 
all-too-common shortcoming of documentation.)

On 1/10/12 5:59 AM, Szabolcs wrote:
> My question if motivated by http://stackoverflow.com/questions/8780068/mathematica-finding-the-conditions-for-the-real-part-of-a-complex-number-to-be
>
> It seems that
>
> Reduce[{ComplexExpand@Re[-1 - Sqrt[a - b] ]<  0, a>  0, b>  0}, {a, b}, Complexes]
>
> will return a different result from
>
> Reduce[{ComplexExpand@Re[-1 - Sqrt[a - b] ]<  0, a>  0, b>  0}, {a, b}]
>
> Also the result of this latter calculation seems incorrect (I didn't expect 'b' to be restricted to be less than 'a')
>
> How does the domain parameter of Reduce work?  Isn't Complexes the default domain?  What changes if we specify Complexes explicitly?  Also, if the result of the second example incorrect (a bug)?
>

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305

• References:
  • The domain parameter of Reduce[]
    • From: Szabolcs <szhorvat@gmail.com>