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:
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)
“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)
“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)