A Borel transform method for locating singularities of Taylor and Fourier series

Walter Pauls, Condensed Matter Theory, Bielefeld, and Observatorie de la Cote Azur, Nice

Complex singularities play an important part in various areas of physics, such as statistical physics, hydrodynamics and quantum mechanics. Power-series (Taylor) expansions, such as low/high temperature expansions for the Ising model, enumerative studies of self-avoiding walks, percolation, and others are frequently used to extract information on the singularities from the coefficients of the corresponding series. Here we propose a new method of analyzing such expansions.

Given a Taylor series with a finite radius of convergence, its Borel transform defines an entire function. A theorem of Polya relates the large distance behavior of the Borel transform in different directions to singularities of the original function. With the help of the new asymptotic interpolation method of van der Hoeven, we show that from the knowledge of hundreds to thousands of Taylor coefficients we can identify very precisely the location of such singularities, as well as their type when they are isolated. This gives us also access to singularities beyond the convergence disk. The method can also be applied to Fourier series of analytic periodic functions and is here tested on various instances constructed from solutions to the Burgers equation. Large precision on scaling exponents (up to twenty accurate digits) can be achieved.