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.
Analysis of superfunctions arose from applications of the evaluation of fractional iterations of functions.
has then been used as the logo of the physics department of the Moscow State University.
[2] Relying on the elegant functional conjugacy theory of Schröder's equation,[3] for his proof, Kneser had constructed the "superfunction" of the exponential map through the corresponding Abel function
There is a book dedicated to superfunctions [4] The recurrence formula of the above preamble can be written as Instead of the last equation, one could write the identity function, and extend the range of definition of the superfunction S to the non-negative integers.
Then, one may posit and extend the range of validity to the integer values larger than −2.
The following extension, for example, is not trivial, because the inverse function may happen to be not defined for some values of
In particular, tetration can be interpreted as superfunction of exponentiation for some real base
For extension to non-integer values of the argument, the superfunction should be defined in a different way.
[5] If the range of holomorphy required is large enough, then the superfunction is expected to be unique, at least in some specific base functions
,[6] but up to 2009, the uniqueness was a conjecture and not a theorem with a formal mathematical proof.
For example, for the transfer function "++", which means unit increment, the superfunction is just addition of a constant.
; and the superfunction approaches unity in the negative direction on the real axis: Similarly, has an iteration orbit In general, the transfer (step) function f(x) need not be an entire function.
An example involving a meromorphic function f reads, Its iteration orbit (superfunction) is on C, the set of complex numbers except for the singularities of the function S. To see this, recall the double angle trigonometric formula Let
The inverse of a superfunction for a suitable argument x can be interpreted as the Abel function, the solution of the Abel equation, and hence The inverse function when defined, is for suitable domains and ranges, when they exist.
At non-negative integer number of iteration, the iterated exponential is an entire function; at non-integer values, it has two branch points, which correspond to the fixed point
Superfunctions, usually the superexponentials, are proposed as a fast-growing function for an upgrade of the floating point representation of numbers in computers.
Such an upgrade would greatly extend the range of huge numbers which are still distinguishable from infinity.
The reconstruction of the superfunction from the transfer function allows to work with relatively thick samples, improving the precision of measurements.
In particular, the transfer function of the similar sample, which is half thinner, could be interpreted as the square root (i.e. half-iteration) of the transfer function of the initial sample.
Similar example is suggested for a nonlinear optical fiber.
[6] It may make sense to characterize the nonlinearities in the attenuation of shock waves in a homogeneous tube.
This could find an application in some advanced muffler, using nonlinear acoustic effects to withdraw the energy of the sound waves without to disturb the flux of the gas.
Again, the analysis of the nonlinear response, i.e. the transfer function, may be boosted with the superfunction.
The square root of this transfer function will characterize the tube of half length.
The mass of a snowball that rolls down a hill can be considered as a function of the path it has already passed.
The mass of the snowball could be measured at the top of the hill and at the bottom, giving the transfer function; then, the mass of the snowball, as a function of the length it passed, is a superfunction.
The operational element may have any origin: it can be realized as an electronic microchip, or a mechanical couple of curvilinear grains, or some asymmetric U-tube filled with different liquids, and so on.
This article incorporates material from the Citizendium article "Superfunction", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL.