# Twisted winding number

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.

• Should $du/u$ be $dz/z$ here? Jan 19, 2021 at 15:13
• Ah yes, will correct appropriately. Thanks Jan 19, 2021 at 15:16
• I misunderstood the question and of course you are correct and I was wrong Jan 19, 2021 at 16:03
• The general question is hopeless if $\chi$ arbitrary as you can take $1$ on an arc, $2$ on the complementary arc and smoothed out at ends and get arbitrary numbers depending on the arc (result is always positive for a circle oriented usually by parametrization) Jan 19, 2021 at 16:07
• @Conrad: Many thanks for your comment. I am not necessarily presuming there’s a closed-form expression for every possible continuous function $\chi$; as may be easily observed, if $\chi$ is constant on $\gamma$, then the result is simply that constant times $n(\gamma,0)$. While I am more interested in the case $\chi(\cdot):=\langle\cdot\rangle^{-2}$, I thought perhaps there might be some results involving some non-trivial hypotheses on $\chi$ and incorporating my case of interest. Jan 19, 2021 at 16:15

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