A function is said to be modular (or "elliptic modular") if it satisfies: 1.f is meromorphic in the upper half-plane H, 2.f(A τ) = f(τ) for every matrix A in the modular group Γ, 3. The Laurent series of f has the form f(τ) = sum_(n = - m)^∞ a(n) e^(2π i n τ) (Apostol 1997, p. 34). Every rational function of Klein's absolute invariant J is a modular function, and every modular function can be expressed as a rational function of J. Modular functions are special cases of modular forms, but not vice versa.
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