analytic continuation

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analytic continuation

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analytic continuation (countable and uncountable, plural analytic continuations)

1. (mathematical analysis) The practice of extending analytic functions.
   • 1968, , Granino A. Korn, Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, 2000, Dover, Unabridged republication, page 206,

     The standard method of analytic continuation starts with a function ${\displaystyle f(z)}$ defined by its power-series expansion (7.5-4) inside some circle ${\displaystyle |z-a|=r}$.

   • 2008, Steven G. Krantz, A Guide to Complex Variables, Mathematical Association of America, page 125:

     Since there are potentially many different ways to carry out this analytic continuation process, there are questions of ambiguity and redundancy.

   • 2014, M. Mursaleen, Applied Summability Methods, Springer, page 31:

     Analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example, in a new region where an infinite series representation in terms of which it is initially defined becomes divergent.

2. (mathematical analysis) An extension of an analytic function which is itself analytic.
   • 1975, S. Smith (translator), L. A. Muraveǐ Asymptotics for the Wave Equation, S. Smith (translator), Valentin P. Michaǐlov (editor), Boundary Value Problems for Differential Equations, Mathematical Association of America, page 107,

     Using (2.6), (2.15)-(2.18) and (2.54) in the same way as in the cases considered earlier, we get that the series (0.14) determines the analytic continuation of the Green function of the second boundary value problem into the domain ${\displaystyle K_{B}\cap \left\{|k|<\beta \right\}}$ with the property (0.23).

   • 1995, Valentin I. Ivanov, Michael K. Trubetskov, Handbook of Conformal Mapping with Computer-Aided Visualization, CRC Press, page 26:

     The set of all the analytic continuations of the given element ${\displaystyle f(z)}$ is called the complete analytic function.