CURRENT Medical Diagnosis and Treatment 2012, Fifty-First by Stephen J. McPhee

By Stephen J. McPhee

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Extra resources for CURRENT Medical Diagnosis and Treatment 2012, Fifty-First Edition

Sample text

44) q func- q function is defined by ⎧ 1−z , ∞ ⎨ 1;q z;q ∞ (1 − q) q (z) ≡ ⎩ 1;q −1 ∞ (q − 1)1−x q (x2) , x;q −1 q ∞ if 0 < |q| < 1, if |q| > 1. 45) Here we deviate from the usual convention q < 1, because we want to work with meromorphic functions of several variables. The reason is that the q-analogue of the Euler reflection formula involves the first Jacobi theta function, which by construction is a complex function, not only real. The simple poles of q are located at z = −n ± 2kπi log q , n, k ∈ N.

Robert Murphy (1806–1843) was a forerunner of George Boole (1815–1864) and Heaviside, who among other things found nice formulas for derivatives in the spirit of Carlitz. In 1854 Arthur Cayley introduced the concept of Cayley table in his article On the theory of groups. Cayley defined a group as a set of objects with a multiplication 38 2 The different languages of q-calculus and a symbolic equation θ n = 1. Blissard was obviously influenced by Cayley’s work, at least in what concerns his notation.

By 1860 two textbooks on finite differences were published in England, one of them by Boole, which covered almost all the theorems that we know now. Heaviside was able to greatly simplify Maxwell’s 20 equations in 20 variables to four equations in two variables. This and other articles about electricity problems, which appeared in 1892–98, were severely criticized for their lack of rigour by contemporary mathematicians. It seems that Heaviside’s contribution to mathematics was underestimated by his contemporaries, since in fact he both discussed formal power series and the rudiments of umbral calculus, which we present in this book.

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