Re: Difficulties with Complex-Modulus Series
- To: mathgroup at smc.vnet.net
- Subject: [mg72751] Re: [mg72712] Difficulties with Complex-Modulus Series
- From: Bob Hanlon <hanlonr at cox.net>
- Date: Wed, 17 Jan 2007 07:32:57 -0500 (EST)
- Reply-to: hanlonr at cox.net
$Version
5.2 for Mac OS X (June 20, 2005)
r = (2*I+x)/(2*I-x);
R=Abs[r];
Series[R,{x,0,4}]
Abs[(2*I + x)/(2*I - x)]
Series[R,{x,0,4},Assumptions->Element[x,Reals]]
SeriesData[x, 0, {-1, I, 1/2, -I/4, -1/8}, 0, 5, 1]
Series[FullSimplify[R,Element[x,Reals]],{x,0,4}]
1
Series[R,{x,0,4},Assumptions->x>=0]
SeriesData[x, 0, {-1, I, 1/2, -I/4, -1/8}, 0, 5, 1]
Series[FullSimplify[R,x>=0],{x,0,4}]
1
Series[ComplexExpand[R],{x,0,4}]
1
Series[R,{x,Infinity,4}]
Abs[(2*I + x)/(2*I - x)]
Series[R,{x,Infinity,4},Assumptions->Element[x,Reals]]
Abs[(2*I + x)/(2*I - x)]
Series[FullSimplify[R,Element[x,Reals]],{x,Infinity,4}]
1
Series[R,{x,Infinity,4},Assumptions->x>=0]
SeriesData[x, Infinity, {-1, -4*I, 8, 16*I, -32}, 0, 5, 1]
Series[FullSimplify[R,x>=0],{x,Infinity,4}]
1
Series[ComplexExpand[R],{x,Infinity,4}]
1
Hence, use assumptions to simplify an expression before using Series
Bob Hanlon
---- carlos at colorado.edu wrote:
> Say I have r = (2*I+x)/(2*I-x), in which x is real and nonnegative.
>
> Series[r,{x,0,4}] and Series[r,{x,Infinity,4}] work as expected.
>
> Introduce now R=Abs[r] and try the same:
>
> Series[R,{x,0,4}] and Series[R,{x,Infinity,4}]
>
> Results are now "contaminated" with Abs'[-1], Abs''[-1], etc,
> I dont understand the presence of those derivatives.
> Anybody can explain the reason? (I teach students that the
> derivative of a constant is zero, but perhaps that has changed
> with the new year) BTW it would be nice to say
>
> Series[R,{x,0,4}, x>=0] or Series[R,{x,0,4}, R>=0] etc
>
> if that would get rid of the garbage, but Mathematica 5
> does not allow Assumptions in Series. Note BTW that R=1 for
> any x, so the R series are in fact trivial to any order.
>