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Superfunction
In mathematics, superfunction is a nonstandard name for an iterated function for complexified continuous iteration index. Roughly, for some function (mathematics), function ''f'' and for some variable ''x'', the superfunction could be defined by the expression :S(z;x) = \underbrace_ . Then, ''S''(''z''; ''x'') can be interpreted as the superfunction of the function ''f''(''x''). Such a definition is valid only for a positive integer index ''z''. The variable ''x'' is often omitted. Much study and many applications of superfunctions employ various ''extensions of these superfunctions to complex and continuous indices''; and the analysis of existence, uniqueness and their evaluation. The Ackermann functions and tetration can be interpreted in terms of superfunctions. History Analysis of superfunctions arose from applications of the evaluation of Iterated_function#Fractional_iterates_and_flows,_and_negative_iterates, fractional iterations of functions. Superfunctions and their inver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abel Function
The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form :f(h(x)) = h(x + 1) or :\alpha(f(x)) = \alpha(x)+1. The forms are equivalent when is invertible. or control the iteration of . Equivalence The second equation can be written :\alpha^(\alpha(f(x))) = \alpha^(\alpha(x)+1)\, . Taking , the equation can be written ::f(\alpha^(y)) = \alpha^(y+1)\, . For a known function , a problem is to solve the functional equation for the function , possibly satisfying additional requirements, such as . The change of variables , for a real parameter , brings Abel's equation into the celebrated Schröder's equation, . The further change into Böttcher's equation, . The Abel equation is a special case of (and easily generalizes to) the translation equation, :\omega( \omega(x,u),v)=\omega(x,u+v) ~, e.g., for \omega(x,1) = f(x), :\omega(x,u) = \alpha^(\alpha(x)+u). (Observe .) The Abel function further provides the canonical coordi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |