- #1
Silviu
- 624
- 11
Hello! Why do the singularities in the Residue Theorem must be isolated? If we have let's say a disk around ##z_0##, ##D_{[z_0,R]}## where all the points are singularities for a function ##f:G \to C## with the disk in region G, but f is holomorphic in ##G-D_{[z_0,R]}##, we can still write f as a Laurent series in ##G-D_{[z_0,R]}## and thus we still have ##\int_{\gamma}f(z)dz = 2\pi i c_{-1}##, with ##\gamma## a picewise, smooth, closed path in ##G-D_{[z_0,R]}##. I know that ##c_{-1}## is called residue only if the singularity is isolated, but letting aside the nomenclature, why can't we use the formula for non-isolated singularities?