Solve an equation perturbatively

Question

I have this equation:

$$T=2 P r-\frac{q^2}{4 \pi r^3}+\frac{1}{4 \pi r},$$

and I want to solve it for $r$ perturbatively. This result should be: $$r=\frac{T}{2 P}-\frac{1}{4 \pi T}+\frac{P \left(8 \pi P q^2-1\right)}{8 \left(\pi ^2 T^3\right)}+\cdots$$

I was reading an article where author did this. How can I do this in Mathematica?

Answer 1

This can be done in such a way.

eq = T == 2*P*r - q^2/(4*Pi*r^3) + 1/(4*Pi*r);
AsymptoticSolve[eq, r, {T, Infinity, 5}][[4]]

{r -> (-(P^2/(4 \[Pi]^2)) + (2 P^3 q^2)/\[Pi])/(2 P T^3) - 1/( 4 \[Pi] T) + T/(2 P)}

The equation eq has four roots, you are interested in the asymptotic of the fourth one as T tends to infinity.