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.

  • $\begingroup$ Should $du/u$ be $dz/z$ here? $\endgroup$
    – Will Sawin
    Jan 19, 2021 at 15:13
  • $\begingroup$ Ah yes, will correct appropriately. Thanks $\endgroup$
    – Jack L.
    Jan 19, 2021 at 15:16
  • $\begingroup$ I misunderstood the question and of course you are correct and I was wrong $\endgroup$
    – Conrad
    Jan 19, 2021 at 16:03
  • $\begingroup$ 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) $\endgroup$
    – Conrad
    Jan 19, 2021 at 16:07
  • $\begingroup$ @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. $\endgroup$
    – Jack L.
    Jan 19, 2021 at 16:15

1 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.


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