Definition
Let
be a formal power series in z.
Define the Borel transform of by
Suppose that the Borel transform converges to an analytic function of t near 0 that can be analytically continued along the positive real axis to a function growing sufficiently slowly that the following integral is well defined (as an improper integral). Then the Borel sum of y is given by
A slightly weaker form of Borel's summation method gives the Borel sum of y as
If the sum exists in this sense then it also exists in the previous sense and is the same, but there are some series that can be summed with the previous method but not with this method (Hardy 1992, 8.5).
Read more about this topic: Borel Summation
Famous quotes containing the word definition:
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (17721834)
“Its a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was mine.”
—Jane Adams (20th century)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)