proof of Cauchy residue theorem
Being holomorphic by Cauchy-Riemann equations the differential form is closed. So by the lemma about closed differential forms on a simple connected domain we know that the integral is equal to if is any curve which is homotopic to . In particular we can consider a curve which turns around the points along small circles and join these small circles with segments. Since the curve follows each segment two times with opposite orientation it is enough to sum the integrals of around the small circles.
So letting be a parameterization of the curve around the point , we have and hence
where is chosen so small that the balls are all disjoint and all contained in the domain . So by linearity, it is enough to prove that for all
Let now be fixed and consider now the Laurent series for in :
so that . We have
Notice now that if we have
while for we have
Hence the result follows.
|Title||proof of Cauchy residue theorem|
|Date of creation||2013-03-22 13:42:04|
|Last modified on||2013-03-22 13:42:04|
|Last modified by||paolini (1187)|