Complex anaylsis, winding number question.


by screwyshrew
Tags: anaylsis, complex, number, winding
screwyshrew
screwyshrew is offline
#1
Mar11-12, 03:37 PM
P: 2
So the in the equation for the winding number/index of a curve

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

where [itex]\gamma[/itex] : [a, b] → ℂ and z is an arbitrary point not on [itex]\gamma[/itex], what exactly does ζ represent?
Phys.Org News Partner Science news on Phys.org
Scientists pinpoint when harmless bacteria became flesh-eating monsters
Asian air pollution affect Pacific Ocean storms
Rocket leak delays space station delivery launch (Update)
mathwonk
mathwonk is offline
#2
Mar11-12, 04:50 PM
Sci Advisor
HW Helper
mathwonk's Avatar
P: 9,421
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
screwyshrew is offline
#3
Mar11-12, 06:32 PM
P: 2
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
mathwonk is offline
#4
Mar11-12, 07:14 PM
Sci Advisor
HW Helper
mathwonk's Avatar
P: 9,421

Complex anaylsis, winding number question.


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. thats the winding number about z = 0 of the unit circle. (times 2pi.i)


Register to reply

Related Discussions
Winding Number iff statement Calculus & Beyond Homework 2
Complex anaylsis question involving liouville's theorem Calculus & Beyond Homework 4
String Theory - Winding Number Beyond the Standard Model 0
Complex log and Winding Number in Pi_1(S^1) Calculus 1
winding number questions Calculus & Beyond Homework 3