How Do Residue Theorems Address Infinity in Complex Analysis?

[nhrock3]

there is a theorem which states that
the resedues of a point inside a certain area
equals minus the resedues outside of the area.

but on the other hand
the resedue in the infinity point equals
minus the sum of the resedues

those two can't coexist together because
infinity is also a point on the inside of an area
so its not letting the first theorem to work
??

 

[Count Iblis]

The sum of all the residues (including the residue at infinity) is zero. This should be easy to prove using the definition of the residue at infinity.