Special functions, which include the trigonometric functions, have been used for centuries. Their role in the solution of differential equations was exploited by Newton and Leibniz, and the subject of special functions has been in continuous development ever since. In just the past thirty years several new special functions and applications have been discovered.This treatise presents an overview of the area of special functions, focusing primarily on the hypergeometric functions and the associated hypergeometric series. It includes both important historical results and recent developments and shows how these arise from several areas of mathematics and mathematical physics. Particular emphasis is placed on formulas that can be used in computation.The book begins with a thorough treatment of the gamma and beta functions, which are essential to understanding hypergeometric functions. Later chapters discuss Bessel functions, orthogonal polynomials and transformations, the Selberg integral and its applications, spherical harmonics, g-series, partitions, and Bailey chains.This clear, authoritative work will be a lasting reference for students and researchers in number theory, algebra, combinatorics, differential equations, mathematical computing, and mathematical physics.
The superoxide radical plays major roles in the neutrophil-medicated acute inflammatory response and in postischemic tissue injury, although the sources and actions of the radical are quite different in these two pathological states. While neutrophils produce superoxide for the primary purpose of aiding in the killing of ingested microbes, a second useful function has evolved. The superoxide released from actively phagocytosing neutrophils serves to attract more neutrophils by reacting with, and activating, a latent chemotactic factor present in plasma. Superoxide dismutase, by preventing the activation of this superoxide-dependent chemotactic factor, exerts potent anti-inflammatory action. During ischemia, energy-starved tissues catabolize ATP to hypoxanthine. Calcium transients in these cells appear to activate a calmodulin regulated protease which attacks the enzyme xanthine dehydrogenase, converting it to a xanthine oxidase capable of superoxide generation. When the tissue is reperfused and reoxygenated, all the necessary components are present (xanthine oxidase, hypoxanthine, and oxygen) to produce a burst of superoxide which results in extensive tissue damage. Ischemic tissues are protected by superoxide dismutase or allupurinol, an inhibitor of xanthine oxidase.
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