= Modular function
{wiki}
A <modular function> for a <finite-index subgroup> $\Gamma\le SL_2(\mathbb Z)$ is a <meromorphic function> on the <complex upper half-plane> that is invariant under $\Gamma$ and is meromorphic at every <modular cusp>:
$$
f(\gamma\tau)=f(\tau)\qquad(\gamma\in\Gamma).
$$
At a <modular cusp> of width $h$, meromorphicity means that the transformed function has a <Laurent series> in the local parameter $q_h=e^{2\pi i\tau/h}$ with only finitely many negative powers. Thus a <modular function> can have poles, in contrast to a <modular form>, which is required to be holomorphic on the half-plane and at the <modular cusps>. The <Klein j-invariant> is a fundamental example for the full <modular group>.
Some sources use the broader expression \i[modular function of weight $k$] for a <meromorphic function> satisfying
$$
f(\gamma\tau)=(c\tau+d)^k f(\tau),\qquad\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix},
$$
with the corresponding meromorphic cusp condition. This is also called a meromorphic <modular form>; the unqualified term <modular function> ordinarily means weight zero. Weight-zero <modular functions> descend to meromorphic functions on the <compactified modular curve>.<Modular Functions and Modular Forms by James S. Milne>[Section 4, definitions of modular functions and modular forms]{ref}
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