In p.312 of & # 39; Rhoades, Robert C., Linear relationships between Poincaré series through weak harmonic forms of Maass. Ramanujan J. 29 (2012), no. 1-3, 311–320 & # 39 ;, the author defines the extended main part at infinity as follows:

A weakly holomorphic modular shape is any meromorphic modular shape whose poles are supported at the cusps. The main part extended into infinity in a weakly holomorphic modular form $ f $ is the polynomial $ P_ {f, infty} in mathbb C (q ^ {- 1}) $ such that $ f (z) -P_ {f, infty} (q ^ {- 1}) = O (e ^ {- epsilon y}) $ how $ y to infty $.

My questions are

- What is the meaning of the extended main part?
- What is the relationship of this definition of extended main part and the & # 39; main part of Laurent's expansion?
- Is the extended main part of the weakly holomorphic modular form uniquely determined?

In addition, the author defines the main part extended at the cusps $ x $ as follows:

Yes $ x $ It is a cusp, the main part extended in $ x $ is the finite sum of terms in the Fourier expansion around $ x $ they have no rapid decline towards $ x $.

I did not understand this sentence, but in my idea the natural definition of the main part extended in $ x $ is the following:

Leave $ sigma in mathrm {SL} _2 ( mathbb R) $ be the satisfying item $ sigma x = infty $. So $ f | _k sigma ^ {- 1} $ has the Fourier expansion in $ q $, where $ q = e (z / h) $, $ h $ is the width of the cusp in $ x $. The main extended part of $ f $ to $ x $ it's a polynomial $ P_ {f, x} (q ^ {- 1}) in mathbb C (q ^ {- 1}) $ such that $ f | _k sigma ^ {- 1} (z) -P_ {f, x} (q ^ {- 1}) = O (e ^ {- epsilon y}) $.

Is my understanding correct?