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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

1 Answers

One Answer

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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