Let $k$,$m$ be non-negative integers. A Jacobi form of weight $k$ and index $m$ is a holomorphic function $f$ on $\mathbb{H} \times \mathbb{C}$ (where $\mathbb{H}$ denotes the upper half plane) satisfying the following two conditions:
i) It satisfies the functional equation $f(\frac{a\tau +b}{c\tau +d}, \frac{z+\lambda \tau + \mu}{c\tau +d})(c\tau +d)^{-k}\exp(2\pi i(\lambda^2\tau + 2\lambda z -\frac{c(z+\lambda \tau + \mu)}{c\tau +d}) = f(\tau,z)$
ii) At the cusp $P1 = [\infty]$, $f$ has a Fourier expansion of the form $f(\tau,z) = \sum_{n \in \mathbb{N}, r \in \mathbb{Z}, 4mn-Nr^2 \geq 0}c(n,r)q^n\zeta ^r$.
My question is the following: If a holomorphic function satisfies the functional equation given in i), then does it have a Fourier expansion $f(\tau,z) = \sum_{n , r \in \mathbb{Z}, }c(n,r)q^n\zeta ^r$? Note that we can prove periodicity in one variable, i.e we have $ f(\tau,z) = f(\tau, z +1)$. Does this imply a Fourier expansion?