# Borel summation method

A method for summing series of functions, proposed by E. Borel . Suppose one is given a series of numbers (*)

let be its partial sums and let be a real number. The series (*) is summable by the Borel method ( -method) to the number if There exists an integral summation method due to Borel. This is the -method: If then one says that the series (*) is summable by the -method to the number . For conditions under which the two methods and are equivalent, cf. . The -method originated in the context of analytic extension of a function regular at a point. Let be regular at the point and let be the set of all its singular points. Draw the segment and the straight line normal to through any point . The set of points on the same side with for each straight line is denoted by ; the boundary of the domain is then called the Borel polygon of the function , while the domain is called its interior domain. The following theorem is valid: The series is summable by the -method in , but not in the domain which is the complement of .