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American Mathematical Monthly - March 2001

 

MARCH 2002


The Curious History of Faà di Bruno’s Formula
by Warren P. Johnson
wjohnson@bates.edu
Faà di Bruno’s formula is by far the best-known answer to the question: What is the mth derivative of a composite function? But it is not the only answer, and probably not the best answer; neither is it really Faà di Bruno’s answer, in the sense that at least three other mathematicians published it before him. We discuss the history of Faà di Bruno type formulas and the connections among them.


 

Laplace’s Integral, the Gamma Function, and Beyond
by Wladimir de Azevedo Pribitkin
wladimir@Princeton.edu, w_pribitkin@msn.com
In 1812 Laplace discovered, in his own words, a "résultat remarquable" that gives an integral formula for the reciprocal of the gamma function. We supply two simple proofs of this fundamental identity, which we exploit for a host of applications. These include an easy proof of the gamma function’s exponential decay in vertical strips and a derivation of summation formulas that are used to capture the Fourier coefficients of automorphic forms. As we demonstrate, these coefficients contain striking arithmetic information, which is indeed remarkable -- way beyond the sense of Laplace.

 

Associativity of the Secant Method
by Sam Northshield
samuel.northshield@plattsburgh.edu
Iterating a function like 1 + 1/x gives a sequence that converges to the Golden Mean but does so at a much slower rate than sequences derived from Newton’s method or the secant method. There is, however, a surprising relation among all these sequences. This relation, easily explained by the use of good notation, is generalized by means of Pascal’s "Mysterium Hexagrammicum." Throughout, we make contact with many areas of mathematics and physics including abstract groups, calculus, continued fractions, differential equations, elliptic curves, Fibonacci numbers, functional equations, fundamental groups, Lie groups, matrices, Möbius transformations, Pi, polynomial approximation, relativity, and resistors.

 

Renewal Systems, Sharp-Eyed Snakes, and Shifts of Finite Type
by Aimee Johnson and Kathleen Madden
aimee@swarthmore.edu, kmadden@drew.edu
Bi-infinite sequences of symbols occur in several different areas of mathematics, including symbolic dynamical systems and coding theory. The question of when two different sets of sequences are "the same" is fundamental. For example, by a simple renaming of the symbols, it is clear that the set of sequences consisting of any arrangement of zeros and ones is "the same" as the set of sequences consisting of any arrangement of a’s and b’s. Consider the set of sequences consisting of symbols {1,2,3,4 } satisfying the requirements that one and four can be followed only by one or three, and that two and three can be followed only by two or four. The fact that this set of sequences in also "the same" is not as obvious! In this paper, we consider examples drawn from two important classes, renewal systems and shifts of finite type. We discuss what it means for two sets of sequences to be the same and, with the help of our sharp-eyed snakes, we investigate several examples. The paper ends with the tantalizing open question of whether every shift

 

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