The residue theorem that states that if
a) $U$ is a simply connected and open subset of the complex plane,
b) $a_1,\dots,a_n$ are finitely many points of $U$,
c) and $f$ is a function which is defined and holomorphic on $U\backslash \{a_1,\dots,a_n \}$,
d) $\gamma$ is a rectifiable and positively oriented curve in $U$ which does not meet any of the $a_k$, and whose start point equals its end point,
then $ \oint_\gamma f(z)\, dz = 2\pi i \sum_{k=1}^n \operatorname{Res}( f, a_k ). $ Question: What is the intuitive explanation of this theorem ?
I appreciate a explanation like geometry ( whit Java Applet ? ). But algebric explanation are welcome. I know the proof of this theorem, I'm just trying to understand the intuition ( if exists ) this theorem.
I'm motivated to get an intuition of this theorem because he gave a wonderful explanation of the intuitive of an part of fundamental theorem of calculus using the properties of the telescopic sums, mean value theorem and Riemann sum whit partition $\{a=x_0
It makes me think that there may be an explanation as elegant as this (but not necessarily following the same reasoning) in understanding the theorem of residues.