Re: Re: branch of (-1)^(1/3)
- To: mathgroup at smc.vnet.net
- Subject: [mg94569] Re: [mg94508] Re: branch of (-1)^(1/3)
- From: Carl Woll <carlw at wolfram.com>
- Date: Tue, 16 Dec 2008 02:37:08 -0500 (EST)
- References: <200812121154.GAA27892@smc.vnet.net> <gi2uug$a6m$1@smc.vnet.net> <200812151244.HAA26236@smc.vnet.net>
slawek wrote: >U¿ytkownik "Carl Woll" <carlw at wolfram.com> napisa³ w wiadomo¶ci >news:gi2uug$a6m$1 at smc.vnet.net... > > >>I assume you mean a simple way to choose the branch of a^(1/3), where a >>is real. If so, you can use: >> >>Root[#^3-a&, 1] >> >> > > >It doesn't work, because this appoach may be used in this way > >In[24]:= (-1)^(1/3) /. a_^(1/3) -> Root[#^3 - a &, 2] >Out[24]= 1/2 (1 - I Sqrt[3]) > >nevertheless is completly unusable in this example > >In[25]:= (-Sin[x])^(1/3) /. a_^(1/3) -> Root[#^3 - a &, 2] >Out[25]= Root[Sin[x] + #1^3 &, 2] > > Why is it unusable? For example, Plot[Root[Sin[x] + #^3 &, 1], {x, -Pi, Pi}] generates a nice plot. Just because the Root object only simplifies when x takes on special values doesn't mean there is anything wrong with it. Carl Woll Wolfram Research >You can see, that Root works only for "numbers" - whereas a simple Sin[x] is >enouch to stop evaluating Root[ ]. > >Obviously, I still can make the calculation by the pencil... and maybe my >old log ruler. Is Mathematica suitable for? > >And there is no help if > >something /. a_^(1/3) -> Abs[a]^(1/3) > >because the ^(1/3) still will fail recognize that the Abs[a] is real and >that most obvious is a real result. (Because there are many things that are >really real - for example taxes - if in any country the tax will be computed >as (income/factor)^(-1/3) it would mean that the tax is imaginary!!! :) ) > >Regarding the numbering of roots: the system is arbitrary, it may be any >permutation and the mathematics will remain the same. >The real - non-real is real: we are in R or in C, quite different sets. >The "branch number" is artifical similary as artifical are plate numbers on >cars. > >slawek > > >
- References:
- branch of (-1)^(1/3)
- From: "slawek" <human@site.pl>
- Re: branch of (-1)^(1/3)
- From: "slawek" <human@site.pl>
- branch of (-1)^(1/3)