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

By Stephen J. McPhee

The number one every year up-to-date textual content in inner medicine!

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Sample text

80). 128) is a daughter nucleus of the q-Vandermonde formula. 273) is a daughter nucleus of the q-Gaussian summation formula. 6 Analogies between q-analysis and analysis For the convenience of the reader we make tables with the relevant concepts exemplified for the relevant sections. In each table the continuous terms are always on the right. 1 and Chapter 6. 3 are somewhat related, because they use the same q-Stirling numbers. 1, Chapter 6, the difference is, for example, in the definition of the q-derivative, which is the same for formal power series.

Certainly these ancient scientists had some notation for trigonometric functions and some of this great research has survived through Latin translations during the renaissance. Georg Purbach (1423–1461), who was born near Linz, has enriched the trigonometry and astronomy with new tables and theorems. He actually combined the chords of Ptolemaios with the sines of the arabs and in that way introduced the first tables of sines with decimals. His student in Vienna Johannes Müller (1436– 1476) continued Purbach’s research and inter alia introduced the tangent in astronomy; Müller called it foecundus (Latin: fruitful) because of its great advantages.

We find a q-analogue of [425, p. 23]: q (α + n) q (α) = {α}n,q . 46) The following equation holds: q (1 + n) = {n}q !. 47) We denote the limit for 0 < |q| < 1 by limq→1− and the limit for |q| > 1 by limq→1+ . 22 1 We have limq→1− q (x) = (x) [343] and limq→1+ For q < 1 the residue at x = −n is [220, p. 17] lim (x + n) x→−n q (x) = q (x) = (1 − q)n+1 . −n; q n log(q −1 ) Introduction (x). 48) n We find by l’Hospital’s rule that limq→1− of this is (−1) n! 42) involving a quotient of q functions evaluated at negative integers.

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