Twisted winding number

Question

Consider the contour integral

$\frac{1}{2\pi i}\oint_\gamma\chi(z)\frac{dz}{z}\,,$

where $\gamma$ is a (not necessarily simple) closed curve lying in $\mathbb{C}\setminus{0}$ and $\chi\colon\mathbb{C}\to\mathbb{R}_{\ge 0}$ 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 $\chi(z)=\langle z\rangle^{-2}$, where $\langle\cdot\rangle:=(1+|\cdot|^2)^{1/2}$ 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.

Answer 1

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

$$ \frac{1}{2\pi i}\int_\gamma \chi(z)\frac{dz}{z}=\frac{1}{\pi}\int_U\bar{\partial}\left(\frac{\chi(z)}{z}\right)d\text{Area}(z)=\frac{1}{\pi}\int_U\left(\frac{\bar{\partial}\chi(z)}{z}\right)d\text{Area}(z)+\chi(0)n(\gamma,0), $$ 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.