This question is about the nature of branch points which arise in certain Cauchy-integral representations of functions of a single complex argument, $z$. The main application is for dispersion relations but we can ignore that connection since this is a straightforwardly mathematical question. (I posted to math.stackexchange but no attention, even with 50 point bounty.)

Suppose we have the following representation: \begin{align} f(z) &= \frac{1}{\pi}\int_0^\infty dx \frac{\Im f(x)}{x-z}, \end{align} where we assume the integral exists for $z=0$. This function has branch-point singularities at $z=0,\infty$ due to the fact that the (constant) endpoints of integration coincide with the pole of the integrand at those points.

The question that I have regarding the order of the branch point is equivalent to understanding the behavior of $f(z)$ as a circuit is traced around the branch point $z=0$.

Suppose I define the "first sheet" as the values $f^{(I)}$ of $f(z)$ where $z=x_0+i\epsilon$ ($\epsilon$ infinitesimal, as usual). I consider a circuit in the $z$-plane as $z=x_0e^{i\theta}$ ($0<\theta<2\pi$). The function is analytic at all points on this trajectory. Once, however, we consider $2\pi < \theta$ we must distort the contour to avoid the singularity in the integrand at $x=x_0$. This effects the analytic continuation of $f(z)$ onto what we'll call the "second sheet". The value $f^{(II)}$ of $f(x_0)$ on the second sheet is \begin{align} f^{(II)}(x_0) &= \frac{1}{\pi}\int_0^\infty dx \frac{\Im f^{(I)}(x)}{x-(x_0-i\epsilon)},\\ &= \frac{1}{\pi}\int_0^\infty dx \frac{\Im f^{(I)}(x)}{x-x_0}-i\,\Im f^{(I)}(x_0). \end{align} This demonstrates that $f(x_0+i\epsilon)-f(x_0-i\epsilon) = 2i\,\Im f(x_0)$ and shows that the integration contour is the branch cut.

We can continue the above procedure, I believe, an arbitrary number of times. So the order of the branch point at $z=0$ appears to be of the logarithmic variety (of infinite order).

My question: Is the branch point that arises from an endpoint singularity always logarithmic? (I suspect there should be way to obtain a square-root type of (or "first order") branch point from this Cauchy representation. But I don't see how to construct it given the above arguments.)


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