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UA396LAG - Lagranges Formal Program Introduction Of the...

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Lagrange’s Formal Program Introduction Of the many failed attempts to secure a foundation for the calculus, probably the most spectacular was the formal program introduced by Lagrange [1] in the late 18th century. Enjoying a wide and justly earned reputation for his application of the new analysis to recreate Newton’s mechanics in a purely analytical form, Lagrange no doubt hoped to do the same for the calculus. At the time, the calculus had come under attack as a discipline with no logical foundation. One approach being pursued by Lagrange and Leonhard Euler was to formally manipulate infinite structures such as series, and derive numerical values consistent with the algebraic operations that resulted. Lagrange therefore attempted to find a complete and consistent approach to the formal representation of real valued functions. The idea was to take the general notion of a real valued function, first introduced by Euler, and define the various operators of the calculus via Taylor’s theorem. Thus if f ( x ) was any function we would first set down its expansion in powers of the variable x : f ( x + a ) = A + B x + C x 2 + from which we could read off various operations associated with f . For example, B is the derivative of f at the point a . Due to Lagrange’s reputation, his formal approach enjoyed a certain vogue, until it was realized that only a special class of functions can be represented in this way. Indeed, many functions relevant to applications of mathematics in the sciences cannot be resolved into a simple series of this form. While the initial high regard for the method was unjustified, so was the subsequent total oblivion for this interesting attempt to find a formal basis for analysis. With the benefit of an additional two centuries of mathematical progress, we might profit from Lagrange’s vision by reviewing this program in the light of modern concepts. Asymptotics If Lagrange had known of Poincare’s formulation of asymptotic series which only appeared later in the 19th century, he could have written: f ( x + a ) A + B x + C x 2 + and greatly expanded the set of functions that could be treated. This statement is simply shorthand for a recursive series of limits as x 0 : f ( x + a ) - A 0 Author: Chappell Brown First created: January , 1996 Last revision: October, 1996 Page: 1 © Copyright 1996 All rights reserved. Permission to copy and distribute for non-profit scholarly purposes granted.
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Lagrange’s Formal Program f ( x + a ) - A - Bx x 0 f ( x + a ) - A - Bx - C x 2 x 2 0 It is significant that these operations can all exist as well defined quantities even though the resulting infinite sum may not converge. The resulting asymptotic expansions are also less restrictive in terms of formal manipulation but still yield unique values for the quantities A , B , C , ... . What has changed is the fact that the mapping from the function f to the quantities A , B , C is no longer a one-to-one correspondence in general, since asymptotic series represent a set of functions rather than uniquely defining a single function via convergence.
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