|
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'' 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）』 ■ウィキペディアで「Branch point」の詳細全文を読む スポンサード リンク
|