Connection With Continued Fractions
One of the most important forms in which an analytic continued fraction can appear is as a regular C-fraction, which is a continued fraction of the form
where the ai ≠ 0 are complex constants, and z is a complex variable.
There is an intimate connection between regular C-fractions and Padé tables with normal approximants along the main diagonal: the "stairstep" sequence of Padé approximants R0,0, R1,0, R1,1, R2,1, R2,2, … is normal if and only if that sequence coincides with the successive convergents of a regular C-fraction. In other words, if the Padé table is normal along the main diagonal, it can be used to construct a regular C-fraction, and if a regular C-fraction representation for the function f(z) exists, then the main diagonal of the Padé table representing f(z) is normal.
Read more about this topic: Padé Table
Famous quotes containing the words connection with, connection and/or continued:
“Morality becomes hypocrisy if it means accepting mothers suffering or dying in connection with unwanted pregnancies and illegal abortions and unwanted children.”
—Gro Harlem Brundtland (b. 1939)
“Much is made of the accelerating brutality of young peoples crimes, but rarely does our concern for dangerous children translate into concern for children in danger. We fail to make the connection between the use of force on children themselves, and violent antisocial behavior, or the connection between watching father batter mother and the child deducing a link between violence and masculinity.”
—Letty Cottin Pogrebin (20th century)
“That, upon the whole, we may conclude that the Christian religion not only was at first attended with miracles, but even at this day cannot be believed by any reasonable person without one. Mere reason is insufficient to convince us of its veracity: And whoever is moved by Faith to assent to it, is conscious of a continued miracle in his own person, which subverts all the principles of his understanding, and gives him a determination to believe what is most contrary to custom and experience.”
—David Hume (17111776)