Branch point: Mathematics, Complex analysis, Multivalued function, Continuous function, Arbitrarily large, Riemann surface, Analytic continuation, ... singularity, Geometric function theory - Softcover

 
9786130621179: Branch point: Mathematics, Complex analysis, Multivalued function, Continuous function, Arbitrarily large, Riemann surface, Analytic continuation, ... singularity, Geometric function theory
Softcover
ISBN 10: 6130621175 ISBN 13: 9786130621179
Publisher: Alphascript Publishing, 2010

Synopsis

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point (Ablowitz & Fokas 2003, p. 46). Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept. Branch points fall into three broad categories: algebraic branch points, transcendental branch points, and logarithmic branch points. Algebraic branch points most commonly arise from functions in which there is an ambiguity in the extraction of a root, such as solving the equation z = w2 for w as a function of z. Here the branch point is the origin, because the analytic continuation of any solution around a closed loop containing the origin will result in a different function: there is non-trivial monodromy. Despite the algebraic branch point, the function w is well-defined as a multiple-valued function and, in an appropriate sense, is continuous at the origin.

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Reseña del editor

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point (Ablowitz & Fokas 2003, p. 46). Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept. Branch points fall into three broad categories: algebraic branch points, transcendental branch points, and logarithmic branch points. Algebraic branch points most commonly arise from functions in which there is an ambiguity in the extraction of a root, such as solving the equation z = w2 for w as a function of z. Here the branch point is the origin, because the analytic continuation of any solution around a closed loop containing the origin will result in a different function: there is non-trivial monodromy. Despite the algebraic branch point, the function w is well-defined as a multiple-valued function and, in an appropriate sense, is continuous at the origin.

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Publisher
Alphascript Publishing
Publication date
2010
Language
English
ISBN 10 
6130621175
ISBN 13 
9786130621179
Binding
Tapa blanda
Number of pages
72
Editor
Miller Frederic P., Vandome Agnes F., McBrewster John