Re: complex numbers
- To: mathgroup at smc.vnet.net
- Subject: [mg105909] Re: [mg105904] complex numbers
- From: Bob Hanlon <hanlonr at cox.net>
- Date: Thu, 24 Dec 2009 20:59:10 -0500 (EST)
- Reply-to: hanlonr at cox.net
Presumably, because
Abs[x + I*y] // LeafCount
8
Sqrt[x^2 + y^2] // LeafCount
11
Bob Hanlon
---- Jon Joseph <josco.jon at gmail.com> wrote:
=============
Thanks Bob. One question on you solutions: Since you declare both x and y to be real in the following
Assuming[{Element[{x, y}, Reals]},
#[{Re[z], Im[z], Abs[z]}] & /@ {Simplify, FullSimplify}]
Why doesn't Abs[z] expand to Sqrt[x^2+y^2] the same way it does in ComplexExpand?
Jon
On Dec 24, 2009, at 7:24 AM, Bob Hanlon wrote:
>
> Clear[x, y, z];
> z = x + I*y;
>
> {Re[z], Im[z], Abs[z]}
>
> {Re[x] - Im[y], Im[x] + Re[y], Abs[x + I*y]}
>
> Use assumptions to target specific variables as Real
>
> Assuming[{Element[{x, y}, Reals]},
> #[{Re[z], Im[z], Abs[z]}] & /@ {Simplify, FullSimplify}]
>
> {{x, y, Abs[x + I*y]}, {x, y, Abs[x + I*y]}}
>
> Assuming[{Element[{x, y}, Reals]},
> #[{Re[z], Im[z], Abs[z]}] & /@ {Simplify, FullSimplify}]
>
> {{x, y, Abs[x + I*y]}, {x, y, Abs[x + I*y]}}
>
> If appropriate (i.e., all variables are Real), use ComplexExpand
>
> {Re[z], Im[z], Abs[z]} // ComplexExpand
>
> {x, y, Sqrt[x^2 + y^2]}
>
> Use TagSet for x and y
>
> x /: Im[x] = 0;
> x /: Re[x] = x;
> y /: Im[y] = 0;
> y /: Re[y] = y;
>
> {Re[z], Im[z], Abs[z]}
>
> {x, y, Abs[x + I*y]}
>
>
> Bob Hanlon
>
> ---- Jon Joseph <josco.jon at gmail.com> wrote:
>
> =============
> All: Something fundamental about complex number initialization I am
> failing to understand. I wish to define a complex number
>
> z=x+ I y where x and y are both reals. Can someone tell me the
> proper way to tell Mathematica that x, y are real so that Re[z] returns
> x and Im[z] returns y? Much appreciated, Jon.
>