Wikipedia example Cauchy's Integral formula I do not really understand the wikipedia example that illustrates the usage of Cauchy's integral formula. enter link description here
The exact point I do not get is, how they argue that one can split this up into two integrals. There argument is, that this is given by Cauchy's integral theorem. But since there are plenty versions of this theorem available I may be missing something, cause I do not see how this could be deduced from the integral theorem
 A: You want to show that the integral over the original circle equals the integral over two smaller circles, one around each pole. This is the same as showing that the sum of the integrals over the three circles is zero, where the big circle is taken counter clockwise and the smaller circles are taken clockwise.
This is the integral on the boundary of a region inside the big circle and outside the two smaller circles.
1) First, draw a diameter through the circle that separates the two poles.
2) Next, notice that the original integral is the sum of the integrals over the semicircles you just created, since the contributions on the diameter cancel.
3) Inside each semicircle is a small circle around a pole. The region between the circle around the pole and the semicircle looks like an annulus with a flat side. You can cut this lopsided annulus into two pieces which are "horseshoe shaped". 
The result is four horseshoe shaped pieces. The sum of the integrals on the boundaries of these pieces gives the original integral. The Cauchy Integral Theorem says that integral on the boundary of each piece is zero, because it's simply connected and the function is analytic on each one. 

The second cut is a useful trick to remember - I've seen it in at least one proof of the existence of Laurent expansions on an annulus.
You can also do this with just one cut, by cutting on a chord of the original circle which goes through both small circles.

