A First Course in Modular Forms
《A First Course in Modular Forms》 · 第 4 页 · Section 1.1「First definitions and examples」，Definition 1.1.2
A function $f:\mathcal H\to\mathbb C$ is a modular form of weight $k$ if it is holomorphic on $\mathcal H$, weakly modular of weight $k$, and holomorphic at $\infty$.