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It is often written in terms of <math>q=\exp(2\pi i z)</math> (the square of the [[nome (mathematics)|nome]]), as:
::<math>f(z)=\sum_{n=-m}^\infty a_n q^n.</math>
This is also referred to as the ''q''-expansion of ''f'' ([[q-expansion principle]]). The coefficients <math>a_n</math> are known as the Fourier coefficients of ''f'', and the number ''m'' is called the order of the pole of ''f'' at i∞. This condition is called "meromorphic at the cusp", meaning that only finitely many negative-''n'' coefficients are non-zero, so the ''q''-expansion is bounded below, guaranteeing that it is meromorphic at ''q'' = 0. <ref>A [[meromorphic]] function can only have a finite number of negative-exponent terms in its Laurent series, its q-expansion. It can only have at most a [[Pole (complex analysis)|pole]] at ''q'' = 0, not an [[essential singularity]] as exp(1/''q'') has.</ref>
Sometimes a weaker definition of modular functions is used – under the alternative definition, it is sufficient that ''f'' be meromorphic in the open upper half-plane and that ''f'' be invariant with respect to a sub-group of the modular group of finite index.<ref>{{Cite book |last1=Chandrasekharan |first1=K. |title=Elliptic functions |publisher=Springer-Verlag |year=1985 |isbn=3-540-15295-4}} p. 15</ref> This is not adhered to in this article.
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