| Modular Function |
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| moduli theory | |
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In Mathematics , modular functions are certain kinds of Mathematical Function s mapping Complex Number s to complex numbers. There are a number of other uses of the term "modular function" as well; see below for details. Formally, a function ''f'' is called modular or a '''modular function''' Iff it satisfies the following properties: # ''f'' is Meromorphic in the open Upper Half-plane ''H''. # For every Matrix ''M'' in the Modular Group Γ , ''f''(''M''τ) = ''f''(τ). # The Laurent Series of ''f'' has the form :: It can be shown that every modular function can be expressed as a Rational Function of Klein's Absolute Invariant ''j''(τ), and that every rational function of ''j''(τ) is a modular function; furthermore, all Analytic modular functions are Modular Form s, although the converse does not hold. If a modular function ''f'' is not identically 0, then it can be shown that the number of zeroes of ''f'' is equal to the number of Pole s of ''f'' in the Closure of the Fundamental Region ''R''Γ. OTHER USES There are a number of other usages of the term ''modular function'', apart from this classical one; for example, in the theory of Haar Measure s, it is a function Δ(''g'') determined by the conjugation action. REFERENCES
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