Point of interest for complex multi-valued functions
In the mathematical field of complex analysis, a branch point of a multivalued function is a point such that if the function is -valued (has values) at that point, all of its neighborhoods contain a point that has more than values.[1] 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 for as a function of . 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 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.[2] In other areas of complex analysis, the unqualified term may also refer to the more general branch points of transcendental type.
^Das, Shantanu (2011), "Fractional Differintegrations Insight Concepts", Functional Fractional Calculus, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 213–269, doi:10.1007/978-3-642-20545-3_5, ISBN 978-3-642-20544-6, retrieved 2022-04-27 (page 6)
In the mathematical field of complex analysis, a branchpoint of a multivalued function is a point such that if the function is n {\displaystyle n} -valued...
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is unramified. The image point y = f ( x ) ∈ Y {\displaystyle y=f(x)\in Y} of a ramification point is called a branchpoint. Let f : X → Y {\displaystyle...
single point x = 0. A bigger barrier is needed in the complex plane, to prevent any closed contour from completely encircling the branchpoint z = 0....
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{C} } because two branches may give different values at a point where both are defined. Compare, for example, the principal branch Log z {\displaystyle...
values. This is the principal branch of the log function. Often it is defined using a capital letter, Log z. BranchpointBranch cut Complex logarithm Riemann...
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parameters including branching angle, distance between nodes or branch points (internode length), and number of branches per branchpoint. Fractal-like patterns...
the 5' splice site of an intron; Splicing factor 1 binds to the intron branchpoint sequence; U2AF1 binds at the 3' splice site of the intron; U2AF2 binds...
separates the principal branch from the two branches W−1 and W1. In all branches Wk with k ≠ 0, there is a branchpoint at z = 0 and a branch cut along the entire...
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functions of a complex variable have branch points. For example, for the nth root and logarithm functions, 0 is a branchpoint; for the arctangent function,...
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primary radicals (at the end of the chain), and tertiary radicals (at a branchpoint) are more stable yet. Each time an ethylene monomer is added, it creates...
process Node (linguistics), a branchpoint in the Tree model, or Node Theory, of language evolution Node (physics), a point along a standing wave where...
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{-7}}} . For a more thorough discussion, see the articles Square root and Branchpoint. As a complex number, the imaginary unit follows all of the rules of...
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