Re: How Mathematica select one of the roots?
- To: mathgroup@smc.vnet.net
- Subject: [mg12318] Re: How Mathematica select one of the roots?
- From: A Hayes <hay@haystack.demon.co.uk>
- Date: Thu, 7 May 1998 18:52:02 -0400
- References: <6imdev$gam@smc.vnet.net>
--
Allan Hayes
msm@smol.carrier.kiev.ua wrote:
> I have some question about Mathematica 3.0.
>
> When Mathematica 3.0 calculate
> (-1.)^(1/3)
> it generate complex value result
> 0.5 + 0.8660254037844385*I.
>
> But when I try to Solve equation
> x^3+1==0
> it generates three different roots.
>
> So, here is the question:
>
> "How Mathematica 3.0 select one of the roots in the first example?"
I believe that Mathematica selcts the principal value of a power:
The principal value of z^p (for non-zero z) is
Exp[p Log[z]]
where Log[z] is the "principal logarithm" of z that is log[Abs[z]] + I
Arg[z]
where log is the "ordinary " logarithm of the positaive real number
Abs[z] and Arg[z] is the "principal argument" of z ( ArgTan[x,y]) that
is the angle, a, in the range (-pi, +pi] (not including -pi) such
that
Exp[I a] = z/Abs[z]
Put another way, for non-zero z,
Log[z] the unique complex number w such that
Exp[w] = z and -pi < Im[w] <= pi and z^a is Exp[a Log[z]]
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