Re: quadratic solution

Today Ken Oliver wrote in sci.math:

> A colleague from my pre-retirement workplace asked me the following:
>
> Solve:
>
>         (5 + sqrt(x))^2 - 9*(5 + sqrt(x) ) + 20 = 0
>
> This was a test question she had used previously asking for a solution over
> the reals.  No problem.
>
> But this year she put this on a test covering quadratics over the complex
> #'s with coeff's possibly in C.
>
> By the way, sqrt() is traditional "check mark-overscore thingy."
>
> This led to needing a solution of sqrt(x) = -1.  A root of course was
> discarded on the real number test, but what about here.
>
> What is the solution to this over the complex numbers?  "What the rules are
> for applying sqrt() to complex radicand" actually seems to be what the
> problem comes down to.
>
> I a little embarassed to say that I don't know the rules for "principle
> values" or branching here.

Depends how you define sqrt(z).

If you make a cut along the +ve real axis
so 0 < arg(z) <= 2*pi and define sqrt(1.exp(i*0)) = 1, then

sqrt(z) = -1 => sqrt(z) = exp(i*pi) => z = exp(2*pi*i) - the other side of
the cut.

Alternatively, you could take -pi < arg(z) <= pi and sqrt(1) = 1; then
sqrt(z) = -1 => arg(z) = 2*pi; no solution.

-- 
P.A.C. Smith
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