Functional Equations in Several Variables (Encyclopedia of Mathematics and its Applications)

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Summary: Aczel and Dhombre's Functional Equations
Review: This and its precursor volume on functional equations by Aczel are the world's greatest books on functioal equations in my opinion. Functional equations are among the most general equations in mathematics, and therefore anybody who wants to use equations however remote from physical sciences should buy and read these books (with the help of a consultant and/or tutor if necessary to explain them in approximately ordinary English). To give the reader an example, logarithms obey log(a times b) = log a + log b. It turns out that the form of this equation, f(a times b) = f(a) + f(b), characterizes logarithms uniquely under certain fairly general conditions. Functional equations like the last one turn out to uniquely characterize whole fields and categories of both commonly and uncommonly used equations ranging from probability and statistics to information theory and entropy and beyond. For example, how many people know that the bell-shaped curve (called the normal or Gaussian distribution) used in grading students on a curve or average has its own functional equation? Or how many people know that computer or cryptographic codes cannot be shorter than Shannon entropy for average codeword length because of its functional equation properties? Aczel and Dhombres cover these and many other topics. The book is published by Cambridge University Press, which is one of the world's best publishing companies, and Aczel is at Waterloo University in Canada, while Dhombres is (at least, until recently) at a French university. Waterloo and most French universities are among the best in the world.