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. >