Logarithmic form: Difference between revisions

Revision as of 09:59, 30 April 2015

Revision as of 04:01, 28 September 2015

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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 ω has only simple poles with integer residues. On higher-dimensional complex manifolds, the [[Poincaré residue]] is used to describe the distinctive behavior of logarithmic forms along poles.

==Holomorphic log complex==