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The domain parameter of Reduce[]


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)?



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