What does it mean for a domain to lie left to a path? I am studying the book "Complex variables and applications" by James Ward Brown, Ruel Vance Churchill and I don't understand what they mean by the domain to lie left to a path.
How can we tell that the domain is left to the path from the drawing ?

 A: Imagine walking along the path in the direction in which it’s oriented. Put your arms out; the left arm points into the region.
A: When you walk in one direction around a simple closed curve, the "inside" of the curve is either always on your left or always on your right. This simply means that the integration along the curves follow the path around the curve so that the interiors are on the left of the direction in which you traverse the curve.
A: It means that if you are walking around the path in the direction shown by the arrow, then the domain will be on your left.
A: The accepted answer is good enough from the perspective of a mathematician visualizing the complex plane containing a path, spread out before her. But that is a human situation. A mathematical definition of "left" in the complex plane is slightly tricky. In going from the negative to the positive part of the real line, the  half plane to the left of our path is the one containing $\mathrm i$. Unfortunately, there is no way of telling which of $\mathrm i$ and$-\mathrm i$ is which; it is an arbitrary choice. As long as we place $\mathrm i$ in what from our own perspective is the "upper" half of the plane,  the orientive conventions of the plane will accord with our native ones. 
