Superfunction

From Citizendium
Revision as of 22:07, 8 December 2008 by imported>Dmitrii Kouznetsov (try to fix problem with math in preamble)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search

Superfunction is smooth exstension of iteration of other function for the case of non-integer number of iterations.

Routgly

Roughly, if

Failed to parse (syntax error): {\displaystyle {S(z) \atop \,} {= {{\underbrace{f\Big (t)\Big}} \atop {z {\rm ~evaluations~ of~ function~}f } }}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {S(z)~=~ \atop {~} {\underbrace{\exp_a\!\Big(\exp_a\!\big(...\exp_a(t) ... )\big)\Big)} \atop ^{z ~\rm exponentials}} <math> ==Definition== For complex numbers <math>~a~} and , such that belongs to some domain ,
superfunction (from to ) of holomorphic function on domain is function , holomorphic on domain , such that

.

Examples

Addition

Chose a complex number and define function with relation . Define function with relation .

Then, function is superfunction ( to ) of function on .

Multiplication

Exponentiation is superfunction (from 1 to ) of function .

Abel function

Inverse of superfunction can be interpreted as the Abel function.

For some domain and some ,,
Abel function (from to ) of function with respect to superfunction on domain is holomorphic function from to such that

The definitionm above does not reuqire that ; although, from properties of holomorphic functions, there should exost some subset such that . In this subset, the Abel function satisfies the Abel equation.

Abel equation

The Abel equation is some equivalent of the recurrent equation

in the definition of the superfunction. However, it may hold for from the reduced domain .


Applications of superfunctions and Abel functions