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I'm reading the book A First Course In Modular Forms and it defines the term weakly modular of weight $k$ as following:

Let $K$ be an integer. A meromorphic function $f:H\rightarrow\mathbb{C}$ is weakly modular of weight $k$ if $$ f(\gamma(\tau))=(c\tau+d)^kf(\tau) $$

for every $\gamma\in$ SL$_2(\mathbb{Z})$ and $\tau\in H$, where $H$ means the upper half complex plane.

My question is, why in definition we use meromorphic function?

I think meromorphic means such function has poles in a set of isolated points. What if let $\tau$ be such a point? Then the RHS of the equation will be $(c\tau+d)^k\infty$, which is undetermined.

Any suggestion?

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A lot is known about non-holomorphic modular forms (search for Maass Forms). Note that there is no trouble with the definition; if f has a pole at $\tau$, then $\gamma(\tau)$ must also be a pole as the factor $(cx+ d)^k$ is certainly holomorphic at $\tau$. – Thom Tyrrell Jan 5 '13 at 5:14
@ThomTyrrell, oh I suddenly understand.... – hxhxhx88 Jan 6 '13 at 1:20

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