In p.312 of 'Rhoades, Robert C., Linear relations among Poincaré series via harmonic weak Maass forms. Ramanujan J. 29 (2012), no. 1-3, 311–320', the author defines the extended principal part at infinity as follows:
A weakly holomorhpic modular form is any meromorhpic modular form whose poles are supported at the cusps. The extended principal part at infinity of a weakly holomorphic modular form $f$ is the polynomial $P_{f,\infty} \in \mathbb C[q^{-1}]$ such that $f(z)-P_{f,\infty}(q^{-1}) = O(e^{-\epsilon y})$ as $y \to \infty$.
My questions are,
- what is the meaning of the extended principal part.
- What's the relation of this definition of extended principal part and the 'principal part of Laurent expansion'?
- Is the extended principal part of weakly holomorphic modular form determined uniquely?
In addition, the author defines the extended principal part at the cusps $x$ as follows:
If $x$ is a cusp, the extended principal part at $x$ is the finite sum of terms in the Fourier expansion around $x$ that do not have rapid decay toward $x$.
I didn't understand this sentence, but in my idea the natural definition of the extended principal part at $x$ is the following:
Let $\sigma \in \mathrm{SL}_2(\mathbb R)$ be an element satisfying $\sigma x = \infty$. Then $f|_k \sigma^{-1}$ has the Fourier expansion in $q$, where $q=e(z/h)$, $h$ is the cusp width at $x$. The extended principal part of $f$ at $x$ is 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 right?
