A Treatise On The Calculus Of Finite Differences(1st Edition)
Authors:
B. Boole
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ISBN: 1440059004, 978-1440059001
Book publisher: Forgotten Books
Book Summary: ( 4 ) CHAPTER II. direct theorems of finite differences. 1. The operation denoted by A is capable of repetition. For the difference of a function of x, being itself a function of x, is subject to operations of the same kind. In accordance with the algebraic notation of indices, the difference of the difference of a function of x, usually called the second difference, is expressed by attaching the index 2 to the symbol A. Thus AAws= AX- In like manner AAX = AX, and generally AA"~X = AX (1), the last member being termed the nth difference of the function ux. If we suppose ux = »3, the successive values of ux with their successive differences of the first, second, and third orders will be represented in the following scheme: Values of X 1 2 3 4 5 6 . 1 8 27 64 125 216 At*. 7 19 37 61 91 ... AX 12 18 24 30... AX 6 6 6... It may be observed that each set of differences may either be formed from the preceding set by successive subtractions in accordance with the Table of Contents
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