I'm really confused about Winding Numbers in Complex Analysis

Hi ~Death~!

It's 2 …

the two upper loops don't go round -i, so you can flatten them out (alternatively, just "snip them off") without changing the winding number …

what's left obviously winds twice (in the same direction)

 

I think for a general answer to the question, you need to know the orientation of the curves. The winding numbers may cancel each other out if the two curves that wind around the point wind in opposite directions.

OWise, like Tiny Tim said, you just need to look at curve segments that form loops that
wind around (0,1). How many do you see.?

Even more precisely, if you knew the parametrization of the curve, you could integrate
around it to find the winding number.

 

The way you've drawn it, there is!

If you start at the origin, go round the blue circle, then go round the green circle, then go round the red circle, and do some of the curve twice, that path has a winding number of 3 (or of 1, if you go round one of the circles the "wrong way").

But the original diagram is presumably intended to indicate that one follows the curve in the only possible smooth way, and each section once only, giving 2.

 

An old EE prof showed us a trick on how to easily determine winding numbers: start from the point you care about (-i) and draw a straight line out from that in any direction. Count the number of CCW "crossings" of that line, and subtract the CW "crossings". That is your winding number. In this case, assuming the curve is smooth, it is either 2 or -2. Orientation matters!

jason