Parameter is usually assumed to be a positive constant. For , the logistic sequence is entire function; as the real part of approaches to minus infinity.
For the special case , the logistic sequence can be expressed in terms of elementary functions; for this case, the iterations are plotted versus for 0.2, 0.5, 0.8, 1, 1.2, In figure at right.
Evaluation of the logistic sequence
The non-integer iterations of the logistic operator can be constructed using the analytic continuation of the logistic sequence. The logistic sequence is function satisfying the recurrent equation
Initially, such an equation was considered for integer values of , see, for example,  , but then it was generalized for complex values . In the simplest case, the logistic sequence allow the asymptotic representation
where , , .. are real coefficients. These coefficients can be found at the substitution of the asymptotic representation to the recurrent equation. In particular,
The series above diferge, but still allow the fast and precise evaluation. For value of such that is not small, the asymptotic representation
can be used for some natural such that is small.
Inverse of the logistic sequence
For the logistic transfer function, the Abel function is the inverse function of the logistic sequence . The Abel function satisfies the Abel equation
This Abel-function can be expressed through the asimptotic representation, inverting that for the Superfunciton:
Again, the series is asymptotic, and if the argument is not small, the Abel function can be evaluated as
for some natural such that .
The inverse function for the logistic operator can be expressed as follows:
Iterations of the logistic operator
As usually, the combination of the superfunction (which is logistic sequence ) and the Abel function (which is , the inverse of the logistic sequence) allows to evaluate the arbitrary (in particular, fractional and even complex) iterations of the logistic transfer function:
This representation is used to plot the non-integer iterates of the logistic operator, shown in the upper right corner of this article. Namely for the case , the representation through the elementary function could be used too. Such a representation is described below.
The logistic sequence is relatively simple superfunction, and in the case , it can be expressed through the elementary function,
In this case, the Abel function
Such a representation follows also from the table of superfunctions .
The combination gives the expression for the iteration of the transfer function:
Such a representation can be simplified, this leads to the expression
In such a way, for , the iterations of the logistic operator, as well as its Superfunction and the Abelfunction can be expressed through the elementary functions. The last expression could be obtained also using the Schroeder function of the logistic operator.
For values , the logistic sequence appears as superfunction of the logistic operator . Together with the Abel function , this allows to evaluate various iterates of the logistic operator. In particular, the square root of the logistic operator can be evaluated, id est, such function that .
In the similar way, the superfunctions and the Abel functions can be evaluated for various transfer functions. One may evaluate the square root of factorial  (used as logo of the Physics Department of the MSU and as part of the logo of TORI  ), and also the , discussed in , and various superfunctions, including the Ackermann functions.
This article had been copypasted and adopted (in particular, $...$ were replaced to <math>...</math>) from .
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