Re: Re: an equation containg radicals

dimmechan at yahoo.com wrote:
> Dear Daniel,
> Thanks for the response.
>
> Reduce did help me a lot.
>>From application of the argument principle I know that there are no
> solutions
> in the complex domain for the restrictions of parameters I have in the
> problem.
> I just have one question.
>
> Can the following command be a verification that mine equation does not
> exhibit
> solutions in the complex domain under the restrictions of parameters?
>
> K[p_] := 1 - 4*(1 - v)*ii^2*p^2*(1 - Sqrt[e^2 - p^2]/Sqrt[a^2 - p^2])
> a = 1/ii;
> eq = K[p] == 0;
> consts = {e -> 1/1000, v -> 3/10, ii -> 10^(-5)};
>
> Reduce[{eq, 0 < e < 1/10, 0 < v < 1, 0 < ii < 1/100} /. consts, p,
> Reals]
> False
>
> Reduce[{eq, 0 < e < 1/10, 0 < v < 1, 0 < ii < 1/100} /. consts, p,
> Complexes]
> False
>
> BTW, why we must mention Real as the third argument?
> Reduce search by deafult in the Real domain as far as I know.
> Am I missing something?
>
> Regards
> Dimitris
> [...]

To follow up, my original response had a problem wherein I cut/pasted
from the wrong In/Out. Should have used:

Reduce[{eq, 0 < e < 1/10, 0 < v < 1, 0 < ii < 1/100}, p, Reals]

without substituting the consts values for parameters {e,v,ii}.

When attempting to get complex-valued solutions for p I think Reduce
runs into trouble, ergo the method will not shed light on that
situation. But it does suffice to show there are no real valued
solutions in the stated neighborhood of parameters.

Daniel Lichtblau
Wolfram Research

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