Borel summation

Borel, then an unknown young man, discovered that his summation method gave the 'right' answer for many classical divergent series. He decided to make a pilgrimage to Stockholm to see Mittag-Leffler, who was the recognized lord of complex analysis. Mittag-Leffler listened politely to what Borel had to say and then, placing his hand upon the complete works by Weierstrass, his teacher, he said in Latin, 'The Master forbids it'.

In mathematics, Borel summation is a summation method for divergent series, introduced by Émile Borel (1899). It is particularly useful for summing divergent asymptotic series, and in some sense gives the best possible sum for such series. There are several variations of this method that are also called Borel summation, and a generalization of it called Mittag-Leffler summation.

There are (at least) three slightly different methods called Borel summation. They differ in which series they can sum, but are consistent, meaning that if two of the methods sum the same series they give the same answer.

Throughout let A(z) denote a formal power series

and define the Borel transform of A to be its corresponding exponential series

Let A_n(z) denote the partial sum

A weak form of Borel's summation method defines the Borel sum of A to be

If this converges at z ∈ C to some function a(z), we say that the weak Borel sum of A converges at z, and write ${\displaystyle {\textstyle \sum }a_{k}z^{k}=a(z)\,({\boldsymbol {wB}})}$.