There is a well known definition of modular form
Let $f:\mathbb{H}\rightarrow\mathbb{C}$, $\Gamma=\Gamma_0(N)$ and $k\in\mathbb{Z}$
(i)$f$ is holomorphic on $\mathbb{H}$
(ii)$f|_k\gamma=f$ for any $\gamma\in\Gamma$
(iii)$f$ is meromorphic(resp holomorphic, vanishing) at all cusps of $\Gamma$
For (iii), this can be stated as following way
For any $\gamma\in SL_2(\mathbb{Z})$, $f|_k\gamma(z)=\sum\limits_{n=n_0}^{\infty}a_nq_h^n$ for some $n_0\in\mathbb{Z}$(resp $n_0=0,n_0\in\mathbb{Z}_{>0})$
My question is that whether (iii) is equivalent to (iii)' or not
(iii)' $f|_k\gamma(z)=O(e^{\epsilon y})$ for some $\epsilon>0$ as $y\rightarrow\infty$(resp $f|_k\gamma(z)=O(1)$, $f|_k\gamma(z)=O(e^{\epsilon y})$ for some $\epsilon<0$ as $y\rightarrow\infty)$
Thank you!
Yes, this is equivalent.
As $f(z) = f(\gamma z)$ for any $\gamma \in \Gamma_0(N)$, we see that $f$ is periodic and has a Fourier expansion (at any cusp, though I stick only to $\infty$ in this answer) $f(z) = \sum_{n \in \mathbb{Z}} a(n) q^n$. Coordinates in $q$ both describe the Fourier expansion and the Riemannian chart near the cusp. Your statements are all immediate from the behavior in $q$ of the $q$-expansion.
For example, meromorphy (at this particular cusp) is equivalent to there existing an $N$ such that $a(n) = 0$ for all $n < N$, which is equivalent to to $\lvert f(q) \rvert \ll q^{N}$ as $q \to 0$ in the Fourier expansion.