
Re: [HM] History of complex numbers
Posted:
Jul 12, 2000 9:24 PM


However, it is possible to define Sqrt[z] for ALL complex numbers z in such a way that Sqrt[1] = I. Even more can be said. For every $n$ in IN and every $z$ in C{0} one can choose, in the set
{r in C : r^n = z},
a "principal value" root[z,n] in such a way that
root[1,2] = Sqrt[1] = I.
The definition is as follows.
(1) If Im[z]=0 and Re[z] >= 0, set root[z,n] = THE nonnegative real number $r$ such that r^n = z.
(2) If $z$ is in C{0}, set root[z,n] = root[Abs[z],n]*(Cos[Arg[z]/n]+I*Sin[Arg[z]/n])
where
Arg[z] = THE $a$ in [Pi,Pi] such that Cos[a]+I*Sin[a] = z/Abs[z].
This particular choice of the argument can be defined explicitly in many ways. For example,
ArcCos[Re[z]/Abs[z]] if Im[z]>=0 Arg[z] =  ArcCos[Re[z]/Abs[z]] if Im[z]<0.
You can test these ideas with any good Computer Algebra System. Try, for example, the following commands written in Mathematica language:
root[z_,n_]:=Module[{a,r,num}, a=Arg[z];r=Power[Abs[z],Power[n,1]]; num=r*(Cos[a/n]+I*Sin[a/n]); FullSimplify[num] ]
In Mathematica, Arg[z] is a builtin function, so you don't need to define it as above. In fact, you can experiment with other builtin functions like Root.  Exercise: Try to locate the principal value in the list
Table[Root[(#^n  z)&,i],{i,1,n}]  Many more interesting things could be added here (about different approaches to "standard" choices in other areas of mathematics), but I don't have time now.
Carlos C\'esar de Ara\'ujo

