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branch point : ウィキペディア英語版
branch point
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. 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'' = ''w''2 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. This is in contrast to transcendental and logarithmic branch points, that is, points at which a multiple-valued function has nontrivial monodromy and an essential singularity. In geometric function theory, unqualified use of the term ''branch point'' typically means the former more restrictive kind: the algebraic branch points. In other areas of complex analysis, the unqualified term may also refer to the more general branch points of transcendental type.
==Algebraic branch points==

Let Ω be a connected open set in the complex plane C and ''ƒ'':Ω → C a holomorphic function. If ''ƒ'' is not constant, then the set of the critical points of ''ƒ'', that is, the zeros of the derivative ''ƒ'''(''z''), has no limit point in Ω. So each critical point ''z''0 of ''ƒ'' lies at the center of a disc ''B''(''z''0,''r'') containing no other critical point of ''ƒ'' in its closure.
Let γ be the boundary of ''B''(''z''0,''r''), taken with its positive orientation. The winding number of ''ƒ''(''γ'') with respect to the point ''ƒ''(''z''0) is a positive integer called the ramification index of ''z''0. If the ramification index is greater than 1, then ''z''0 is called a ramification point of ''ƒ'', and the corresponding critical value ''ƒ''(''z''0) is called an (algebraic) branch point. Equivalently, ''z''0 is a ramification point if there exists a holomorphic function φ defined in a neighborhood of ''z''0 such that ''ƒ''(''z'') = φ(''z'')(''z'' − ''z''0)''k'' for some positive integer ''k'' > 1.
Typically, one is not interested in ''ƒ'' itself, but in its inverse function. However, the inverse of a holomorphic function in the neighborhood of a ramification point does not properly exist, and so one is forced to define it in a multiple-valued sense as a global analytic function. It is common to abuse language and refers to a branch point ''w''0 = ''ƒ''(''z''0) of ''ƒ'' as a branch point of the global analytic function ''ƒ''−1. More general definitions of branch points are possible for other kinds of multiple-valued global analytic functions, such as those that are defined implicitly. A unifying framework for dealing with such examples is supplied in the language of Riemann surfaces below. In particular, in this more general picture, poles of order greater than 1 can also be considered ramification points.
In terms of the inverse global analytic function ''ƒ''−1, branch points are those points around which there is nontrivial monodromy. For example, the function ''ƒ''(''z'') = ''z''2 has a ramification point at ''z''0 = 0. The inverse function is the square root ''ƒ''−1(''w'') = ''w''1/2, which has a branch point at ''w''0 = 0. Indeed, going around the closed loop ''w'' = ''e''i''θ'', one starts at θ = 0 and ''e''i0/2 = 1. But after going around the loop to θ = 2π, one has ''e''2πi/2 = −1. Thus there is monodromy around this loop enclosing the origin.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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