### 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.