Residue Currents in one Dimension Different Approaches

Abstract

Let f be an holomorphic function in a domain U ⊂ ℂ.We assume that f ≢ 0, so that the zero set of f is a discrete subset of U. Let K be a relatively compact subset of U and Φ ∈ D(U) such that supp(Φ) ⊂ K, our first goal is to study the expression:

$$ I\left( {\O,\varepsilon } \right) = \frac{1}{{2\pi i}}\int\limits_{\left| f \right| = \varepsilon } {\frac{{\O \left( \zeta \right)}}{{f\left( \zeta \right)}}} d\zeta $$

(1.1)

and more precisely what happens if є tends to 0.