Borel Summation - The Borel Polygon

The Borel Polygon

The region where the power series of an analytic function is Borel summable was described as follows by Borel and Phragmen (Sansone & Gerretsen 1960, 8.3).

If y is a power series that converges in some neighborhood of the origin then it has a Borel sum at some point z if it can be analytically continued to a disc with diameter 0z. Conversely if the function can be analytically continued to the disc with diameter 0z then it is Borel summable at z.

The set of points z such that the function can be analytically continued to the interior of the disk with diameter 0z is a polygon when the function has only a finite number of singularities, called the Borel polygon. Its edges pass through the singular points and are orthogonal to lines joining the singular points to 0.