In mathematics, a Borel summation is a generalisation of the usual notion of summation of a series. In particular it gives a definition of a quantity that in many ways behaves formally like the sum, even if the series is in fact a divergent series. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a divergent series is a series that does not converge. ...

Definition

Let

be a formal power series in z.

Define the Borel transform of y by

.

Suppose that

1. has a nonzero radius of convergence as a function of t
2. can be analytically continued to a function on all of the positive real line
3. grows at most exponentially along the positive real line

Then the Borel sum of y is given by the Laplace transform of . This function is guaranteed to exist by condition (3) above. In mathematics and in particular, in functional analysis, the Laplace transform of a function f(t) defined for all real numbers t ≥ 0 is the function F(s), defined by: The lower limit of 0− is short notation to mean and assures the inclusion of the entire dirac delta function...

Discussion

The Borel sum of a series is the Laplace transform of the sum of the term-by-term inverse Laplace transform of the original series. If the Laplace transform of an infinite series were equal to the sum of its term-by-term Laplace transform then the Borel sum would be equal to the usual sum. The Borel sum is defined in many situations where the sum isn't defined. Speaking nonrigorously, it allows us to attach a meaning to the 'sum' of certain types of divergent series. Borel summation is an example of a moment constant method for summing series.

Applications

Borel summation finds application in perturbation theory where physicists frequently require the sum of a series even though it is divergent. In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. ...