First of all, I want to apologize for the names I'm going to use on this wiki, because many of them probably have different names when written in books. We'll suppose that is an open set throughout, unless otherwise stated, and means is a holomorphic function in . We are going to deal with integrals, series, Bernouilli numbers, Riemann zeta function, and many interesting problems, as well as many theories. It's very important to tell everything is very joined and connected inside of complex analysis, so we'll use much knowledge of complex analysis and all the branches of mathematics.
Relevant wiki: Contour Integration
Definition of Winding Number
If is a closed piecewise smooth path and , the winding number of with respect to is Intuitively, is the number of times loops around anticlockwise.
If for its winding number about is
Nevertheless, we can't say because is not closed.
The winding number of a path with respect to is constant in the connected component of containing (or not) . This is, the winding number is 0 if doesn't turn around and it's constant if turns around . (Think about the winding number as the number of turns around...)
Example 2 (Generalization of Example 1)
If for then