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:<math>\omega = \frac{df}{f} =\left(\frac{m}{z} + \frac{g'(z)}{g(z)}\right)dz</math>
for some [[meromorphic function]] (resp. [[rational function]]) <math> f(z) = z^mg(z) </math>, where ''g'' is holomorphic and non-vanishing at 0, and ''m'' is the order of ''f'' at ''0''.. That is, for some [[open covering]], there are local representations of this differential form as a [[logarithmic derivative]] (modified slightly with the [[exterior derivative]] ''d'' in place of the usual [[differential operator]] ''d/dz''). Observe that <math> \omega </math> has only simple poles with integer residues. On higher dimensional complex manifolds, the [[Poincaré residue]] is used to describe their distinctive behavior along poles.
==Holomorphic Log Complex ==
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:<math> \Omega_X^k(\log D)_p = \bigwedge^k_{j=1} \Omega_X^1(\log D)_p </math>.
Some authors, e.g. <ref name = "foo2">Phillip A. Griffiths; Joseph Harris (1979). Principles of Algebraic Geometry. Wiley. </ref>, use the term ''log complex'' to refer to the holomorphic log complex corresponding to a divisor with normal crossings.
===Higher Dimensional Example===
Consider a punctured elliptic curve, given as the locus ''D'' of complex points <math> (x,y) </math> satisfying <math> g(x,y) = y^2 - f(x) = 0 </math>, where <math>f(x) = x(x-1)(x-\lambda) </math> and <math> \lambda\neq 0,1 </math> is a complex number. Then ''D'' is a smooth irreducible hypersurface in <math>\mathbb{C}^2</math> and, in particular, a divisor with simple normal crossings. There is a meromorphic two-form on <math>\mathbb{C}^2 </math>
:<math> \omega = \frac{dx\wedge dy}{g(x,y)} </math>
which has a simple pole along ''D''. The Poincaré residue of <math>\omega </math> along ''D'' is given by the holomorphic one-form
:<math> \text{Res}_D(\omega) = \frac{dx}{\partial g/\partial y}|_D = \frac{1}{2}\frac{dx}{y}|_D </math>.
=== Hodge Theory ===
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