Complex anaylsis, winding number question.

[screwyshrew]

So the in the equation for the winding number/index of a curve

I(\gamma, z) = \frac{1}{2i\pi} \int\gamma \frac{1}{ζ-z}dζ

where \gamma : [a, b] → ℂ and z is an arbitrary point not on \gamma, what exactly does ζ represent?

 

[mathwonk]

It represents an arbitrary point of the curve gamma. I.e. it is a bound or "dummy" variable for any point of gamma. a fuller notation would be the integral over all zeta on gamma. But the fact that it appears after the "d" also tells you it is the variable of integration, and is usual.

 

[screwyshrew]

This is possibly a dumb question (please bear with me): the integration would be taken from a to be, correct? Could you post an example of this equation in use, please?

 

[mathwonk]

yes, after parametrizing the curve gamma by an interval [a.b], then the integral is pulled back to that interval and THEN taken from a to b.

e.g. just integrate dw/w over the unit circle, parametrizing it by cos(t) +i sin(t) for the interval [0,1]. try it. that's the winding number about z = 0 of the unit circle. (times 2pi.i)