In mathematics, superfunction is a nonstandard name for an iterated function for complexified continuous iteration index. Roughly, for some function f and for some variable x, the superfunction could be defined by the expression
S ( z ; x ) = f ( f ( … f ( x ) … ) ) ⏟ z evaluations of the function f . {\displaystyle S(z;x)=\underbrace {f{\Big (}f{\big (}\dots f(x)\dots {\big )}{\Big )}} _{z{\text{ evaluations of the function}}\,f}.}
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 fractional iterations of functions. Superfunctions and their inverses allow evaluation of not only the first negative power of a function (inverse function), but also of any real and even complex iterate of that function. Historically, an early function of this kind considered was exp {\displaystyle {\sqrt {\exp }}} ; the function ! {\displaystyle {\sqrt {\,!\;}}} has then been used as the logo of the physics department of the Moscow State University. At that time, these investigators did not have computational access for the evaluation of such functions, but the function exp {\displaystyle {\sqrt {\exp }}} was luckier than ! {\displaystyle {\sqrt {\,!\;}}} : at the very least, the existence of the holomorphic function
φ {\displaystyle \varphi } such that φ ( φ ( u ) ) = exp ( u ) {\displaystyle \varphi (\varphi (u))=\exp(u)} had been demonstrated in 1950 by Hellmuth Kneser. Relying on the elegant functional conjugacy theory of Schröder's equation, for his proof, Kneser had constructed the "superfunction" of the exponential map through the corresponding Abel function X {\displaystyle {\mathcal {X}}} , satisfying the related Abel equation
X ( exp ( u ) ) = X ( u ) + 1. {\displaystyle {\mathcal {X}}(\exp(u))={\mathcal {X}}(u)+1.\ }
so that X ( S ( z ; u ) ) = X ( u ) + z {\displaystyle {\mathcal {X}}(S(z;u))={\mathcal {X}}(u)+z\ } . The inverse function Kneser found,
S ( z ; u ) = X − 1 ( z + X ( u ) ) {\displaystyle S(z;u)={\mathcal {X}}^{-1}(z+{\mathcal {X}}(u))}
is an entire super-exponential, although it is not real on the real axis; it cannot be interpreted as tetrational, because the condition S ( 0 ; x ) = x {\displaystyle S(0;x)=x} cannot be realized for the entire super-exponential. The real exp {\displaystyle {\sqrt {\exp }}} can be constructed with the tetrational (which is also a superexponential); while the real ! {\displaystyle {\sqrt {\,!\;}}} can be constructed with the superfactorial. There is a book dedicated to superfunctions.
Extensions The recurrence formula of the above preamble can be written as
S ( z + 1 ; x ) = f ( S ( z ; x ) ) ∀ z ∈ N : z > 0 {\displaystyle S(z+1;x)=f(S(z;x))~~~~~~~~\forall z\in \mathbb {N} :z>0}
S ( 1 ) = f ( x ) . {\displaystyle S(1)=f(x).}
Instead of the last equation, one could write the identity function,
S ( 0 ) = x , {\displaystyle S(0)=x~,}
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