Series expansion using complex variables

Question

Suppose I have a function

F[r_]:= A/r + B/r^2 + Conjugate[C]/r^3 

with A, B, C being complex numbers and r being a real variable. I would like to series expand

Re[F[r]^3]

in powers of inverse r. How do I tell mathematica to treat A, B, and C as complex numbers while r as a real variable? Can I do this when defining the function F[r]? Can I do this while using Series[] ? Which is the shortest way to do this?

Answer 1

Letting

F[r_]:= A/r + B/r^2 + Conjugate[C]/r^3

here are two ways:

Series[ComplexExpand[Re[F[r]^3], {A, B, C}], {r, ∞, 4}] // FullSimplify
   (A^3 + Conjugate[A]^3)/(2 r^3) + Re[-3 Im[A] (B Im[A] + 2 A Im[B]) +
   3 B Re[A]^2]/r^4 + O[r, ∞]^5

if you only want a few terms, or

Collect[ComplexExpand[Re[F[r]^3], {A, B, C}] /. r -> 1/u, u, FullSimplify] /. u -> 1/r

if you want the full expansion.