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A normalized Series test


Adding to the previous post, my 5.0 result with Normal[] is

rho=(x+a*I)/(x-a*I); R=Abs[rho];
s=Normal[Series[R,{x,0,4}]];
Print[FullSimplify[s,a>=0&&x>=0]//InputForm];

(3*a^4 - (6*I)*a^3*x - 6*a^2*x^2*(1 + Derivative[2][Abs][-1]) + (
        2*I)*a*x^3*(3 + 6*Derivative[2][
                  Abs][-1] - 2*Derivative[3][Abs][-1]) + 2*x^4*(
        3 + 9*Derivative[2][Abs][-1] - 6*Derivative[
                    3][Abs][-1] + Derivative[4][Abs][-1]))/(3*a^4)

Verification that indeed R=1:

Print[FullSimplify[R,a>=0&&x>=0]//InputForm];  (* Simplify fails *)
1
Print[ComplexExpand[R]//InputForm];
1

Interestingly, I ran this script on 2.2 (which still runs under
Classic). The answer is also messy but the Abs argument is +1.


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