Infinite compositions of analytic functions (original) (raw)
In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually be expanded directly as infinite compositions. In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions. Complex dynamics offers another venue for iteration of systems of functions rather than a single function. For infinite compositions of a single function see Iterated function. For compositions of a finite number of functions, useful in fractal theory, see Iterated function system.
| Property | Value |
|---|---|
| dbo:abstract | In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually be expanded directly as infinite compositions. In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions. Complex dynamics offers another venue for iteration of systems of functions rather than a single function. For infinite compositions of a single function see Iterated function. For compositions of a finite number of functions, useful in fractal theory, see Iterated function system. Although the title of this article specifies analytic functions, there are results for more general functions of a complex variable as well. (en) |
| dbo:thumbnail | wiki-commons:Special:FilePath/Reproductive_universe.jpg?width=300 |
| dbo:wikiPageExternalLink | https://www.coloradomesa.edu/math-stat/catcf/papers/primerinfcompcomplexfcns.pdf http://comet.lehman.cuny.edu/keenl/blochconstantsfinalversion.pdf http://comet.lehman.cuny.edu/keenl/forwarditer.pdf |
| dbo:wikiPageID | 34796035 (xsd:integer) |
| dbo:wikiPageLength | 25916 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID | 1123107793 (xsd:integer) |
| dbo:wikiPageWikiLink | dbc:Analytic_functions dbr:Attractive_fixed_point dbr:Complex_analysis dbr:Analytic_function dbc:Emergence dbr:Entire_function dbr:Function_composition dbr:Generalized_continued_fraction dbr:Möbius_transformation dbr:Contraction_mapping dbr:Complex_dynamics dbr:File:Self-generating_series3.jpg dbr:Active_contour_model dbr:Fixed_point_(mathematics) dbr:Fractal dbr:Iterated_function_system dbr:Product_(mathematics) dbc:Fixed-point_theorems dbr:Iterated_function dbr:File:Continued_fraction1.jpg dbc:Algorithmic_art dbc:Complex_analysis dbr:Zeno's_paradoxes dbr:Series_(mathematics) dbr:Euler's_continued_fraction_formula dbr:Euler_method dbr:Convergence_(mathematics) dbr:File:Contours_in_the_vector_field_f(z)_=_-Cos(z).jpg dbr:File:Diminishing_returns.jpg dbr:File:Dream_of_Gold.jpg dbr:File:Infinite_Brooch.jpg dbr:File:Metropolis_at_30K.jpg dbr:File:Picasso's_Universe.jpg dbr:File:Reproductive_universe.jpg dbr:File:Virtual_tunnels.jpg |
| dbp:mathStatement | {Gn} converges uniformly on compact subsets of S to γ ∈ Ω if and only if the sequence of fixed points {γn} of the {fn} converges to γ. (en) If fn → f where f is parabolic with fixed point γ. Let the fixed-points of the {fn} be {γn} and {βn}. If then Fn → λ, a constant in the extended complex plane, for all z. (en) If an ≡ 1, then Fn → F is entire. (en) Suppose where there exist such that and implies and Furthermore, suppose and Then for (en) Set ε'n = suppose there exists non-negative δ'n, M1, M2, R such that the following holds: Then Gn → G is analytic for < R. Convergence is uniform on compact subsets of {z : < R}. (en) Let f be analytic in a simply-connected region S and continuous on the closure of S. Suppose f is a bounded set contained in S. Then for all z in there exists an attractive fixed point α of f in S such that: (en) {Fn} converges uniformly on compact subsets of S to a constant function F = λ. (en) Suppose is a simply connected compact subset of and let be a family of functions that satisfies Define: Then uniformly on If is the unique fixed point of then uniformly on if and only if . (en) On the set of convergence of a sequence {Fn} of non-singular LFTs, the limit function is either: a non-singular LFT, a function taking on two distinct values, or a constant. In , the sequence converges everywhere in the extended plane. In , the sequence converges either everywhere, and to the same value everywhere except at one point, or it converges at only two points. Case can occur with every possible set of convergence. (en) Suppose where there exist such that implies Furthermore, suppose and Then for (en) Let φ be analytic in S = {z : < R} for all t in [0, 1] and continuous in t. Set If ≤ r < R for ζ ∈ S and t ∈ [0, 1], then has a unique solution, α in S, with (en) If {Fn} converges to an LFT, then fn converge to the identity function f = z. (en) If fn → f and all functions are hyperbolic or loxodromic Möbius transformations, then Fn → λ, a constant, for all , where {βn} are the repulsive fixed points of the {fn}. (en) |
| dbp:name | Theorem (en) Backward Compositions Theorem (en) Contraction Theorem for Analytic Functions (en) Forward Compositions Theorem (en) Theorem E1 (en) Theorem E2 (en) Theorem FP2 (en) Theorem GF3 (en) Theorem GF4 (en) Theorem LFT1 (en) Theorem LFT2 (en) Theorem LFT3 (en) Theorem LFT4 (en) |
| dbp:wikiPageUsesTemplate | dbt:= dbt:Cite_journal dbt:Math dbt:Overline dbt:Portal dbt:Reflist dbt:Short_description dbt:Abs dbt:Math_theorem |
| dct:subject | dbc:Analytic_functions dbc:Emergence dbc:Fixed-point_theorems dbc:Algorithmic_art dbc:Complex_analysis |
| rdf:type | yago:WikicatAnalyticFunctions yago:Abstraction100002137 yago:Communication100033020 yago:Function113783816 yago:MathematicalRelation113783581 yago:Message106598915 yago:Proposition106750804 yago:Relation100031921 yago:Statement106722453 yago:Theorem106752293 yago:WikicatFixed-pointTheorems |
| rdfs:comment | In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually be expanded directly as infinite compositions. In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions. Complex dynamics offers another venue for iteration of systems of functions rather than a single function. For infinite compositions of a single function see Iterated function. For compositions of a finite number of functions, useful in fractal theory, see Iterated function system. (en) |
| rdfs:label | Infinite compositions of analytic functions (en) |
| owl:sameAs | freebase:Infinite compositions of analytic functions wikidata:Infinite compositions of analytic functions https://global.dbpedia.org/id/4nBRb yago-res:Infinite compositions of analytic functions |
| prov:wasDerivedFrom | wikipedia-en:Infinite_compositions_of_analytic_functions?oldid=1123107793&ns=0 |
| foaf:depiction | wiki-commons:Special:FilePath/-Cos(z).jpg wiki-commons:Special:FilePath/Continued_fraction1.jpg wiki-commons:Special:FilePath/Diminishing_returns.jpg wiki-commons:Special:FilePath/Dream_of_Gold.jpg wiki-commons:Special:FilePath/Infinite_Brooch.jpg wiki-commons:Special:FilePath/Metropolis_at_30K.jpg wiki-commons:Special:FilePath/Picasso's_Universe.jpg wiki-commons:Special:FilePath/Reproductive_universe.jpg wiki-commons:Special:FilePath/Self-generating_series3.jpg wiki-commons:Special:FilePath/Virtual_tunnels.jpg |
| foaf:isPrimaryTopicOf | wikipedia-en:Infinite_compositions_of_analytic_functions |
| is dbo:wikiPageWikiLink of | dbr:List_of_complex_analysis_topics dbr:Algorithmic_art dbr:Dynamical_system dbr:Infinite_expression dbr:Analytic_function dbr:Low-complexity_art dbr:Function_composition dbr:Generalized_continued_fraction dbr:Möbius_transformation dbr:Complex_dynamics dbr:Abel_equation dbr:Fixed-point_iteration dbr:Fixed_point_(mathematics) dbr:Flow_(mathematics) dbr:Banach_fixed-point_theorem dbr:Brouwer_fixed-point_theorem dbr:Fractal_art dbr:Iterated_function_system dbr:Iteration dbr:Recursion dbr:Iterated_function dbr:Series_(mathematics) |
| is foaf:primaryTopic of | wikipedia-en:Infinite_compositions_of_analytic_functions |
