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:
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“It’s 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)