# Twisted winding number

Consider the contour integral

$$frac{1}{2pi i}oint_gammachi(z)frac{dz}{z},,$$

where $$gamma$$ is a (not necessarily simple) closed curve lying in $$mathbb{C}setminus{0}$$ and $$chicolonmathbb{C}tomathbb{R}_{ge 0}$$ is a continuous function. My question is

Are there special/generic hypotheses on $$chi$$ that allow for closed-form expressions of the contour integral above?

Of course, in the trivial case of a constant function $$chi$$, the integral is simply the constant time the winding $$n(gamma,0)$$ of $$gamma$$ about the origin. I suspect that the integral will be a (non-negative real) multiple of $$n(gamma,0)$$ but I am unable to figure out how to proceed or determine this factor, if so.

I am particularly interested in finding a closed form expression when $$chi(z)=langle zrangle^{-2}$$, where $$langlecdotrangle:=(1+|cdot|^2)^{1/2}$$ is the Japanese bracket, which has arisen in a harmonic analysis context. If this specific case is known in the literature or tractable, I would be glad for a reference or a proof.

MathOverflow Asked by Jack L. on February 9, 2021

The integral over the curve can be reduced to the integral over the region bounded by the curve using Green's formula:

$$frac{1}{2pi i}int_gamma chi(z)frac{dz}{z}=frac{1}{pi}int_Ubar{partial}left(frac{chi(z)}{z}right)dtext{Area}(z)=frac{1}{pi}int_Uleft(frac{bar{partial}chi(z)}{z}right)dtext{Area}(z)+chi(0)n(gamma,0),$$ in your notation. If your curve is non-simple, then you need to account for multiplicities: count $$k$$ times the integral over all $$z$$ such that $$n(gamma,z)=k$$. Apart from that, not much can be said.

Answered by Kostya_I on February 9, 2021

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