A paper I am reading defines a variable $\theta$ in terms of another variable $\phi$ as an expansion in $u$, where $u$ is understood to be small: $$\theta=\phi-u^2\sin\phi+\mathcal{O}(u^4).$$ They then go on to "invert this relation in the small $u$ limit" and use it to eliminate $\phi$ from another equation. How can I do this in Mathematica?
If I try the obvious guess
Theta[Phi] := Phi - u^2 Sin[Phi]
InverseFunction[Theta][Phi]
The output is just Theta^(-1)[Phi]
.
If I try
AsymptoticSolve[Phi-u^2 Sin[Phi] == Theta, Phi, {u, 0, 1}]
I get AsymptoticSolve[Phi-u^2 Sin[Phi] == Theta, Phi, {u, 0, 1}]
as the output.
Clearly the authors of this paper have managed to find a way to invert this relation, so I am wondering if anyone has any ideas on how to do so. Thanks.