08/10/2007, 08:39 PM
Introduction
Definition: An analytic function with a parabolic rationally neutral fixed-point is a function f(x) that is complex-analytic in x, and satisfies
Notation: We use
Examples of analytic functions with a parabolic rationally neutral fixed-point include: the sine function sin(x)
Iterate-Derivative Matrices
Definition: An iterate-derivative matrix (IDM) of f(x) about
As you can see, the name stems from the iterate
Using Lagrange interpolation, all columns can be turned into finite polynomials:
- Column 0:
- Column 1: 1
- Column 2:
- Column 3:
- Column 4:
One way of doing parabolic iteration is simply to use the IDM and Lagrange interpolation. Since you know that column k is going to be a polynomial in j of degree
which expresses the continuous iteration of an analytic function with a parabolic rationally neutral fixed-point. For this formula to apply to continuous iteration and not just discrete iteration, the gamma function must be used to evaluate the binomial coefficient, or expand it to a polynomial first. Notice that we obtained this formula using the IDM and Lagrange interpolation. We will compare this with other formulas we will derive later.
Power-Derivative Matrices
Definition: A power-derivative matrix (PDM) of f(x) about
The PDM about
As you can see, the PDM turns composition into matrix multiplication, and iteration into matrix powers. This is the key to understanding PDMs. Once this is understood, one can begin thinking in linear algebra instead of function theory, for they become one and the same.
History: Some variants of power-derivative matrices (PDMs) are also known as Bell matrices (if
History: According to Bennet, the first use of PDMs for continuous iteration was by Koch. Kowalski et.al. call the more general form a Carleman matrix, and along with Aldrovandi et.al. refer to the special case when
For an example of a PDM for this kind of function, using
and an example of the simplified case when the fixed-point is zero
As you can see, the PDM for
Another way of finding the continuous iteration of this special kind of function is using PDMs and matrix powers. Since this matrix represents increasing powers of a function and its derivatives, we only need the first power of the function to find its coefficients, which corresponds to the first row (not the zeroth). Putting it all together (using
just like with the IDM method, only with
Double-Binomial Expansions
The first to devote a full-length paper to this method was S. C. Woon. The expansion is based on the binomial theorem, and as such, is easy to understand. Woon originally described his method in terms of the iteration of operators, but here we describe it in terms of functions:
This brings us full circle, since the last part of the above expression is exactly the definition of an IDM. These three methods are very similar and inter-connected, and as such, are all tools of parabolic iteration.
References
R. Aldrovandi, Special Matrices of Mathematical Physics, World Scientific, 2001.
R. Aldrovandi and L.P. Freitas, Continuous Iteration of Dynamical Maps, Journal of Math. Phys. 39, 5324, 1998.
E. T. Bell, The Iterated Exponential Integers, The Annals of Mathematics, 2nd Ser., Vol. 39, No. 3. (Jul., 193

T. Carleman, Acta Mathematica 59, 63, 1932.
D. Geisler, http://Tetration.org, (accessed: 2006).
H. v. Koch, Bil. t. Sv. Vet. Ak. Hand. 1, Math. 25, m'em. no. 5, pp. 1-24, (1900).
H. Trappmann, Arborescent Numbers: Higher Arithmetic Operations and Division Trees, http://blafoo.de/tree-aoc/main1176.pdf, (accessed: 2006).
H. Trappmann and I. Dahl, Tetration extended to real exponents, sci.math.research, http://mathforum.org/kb/message.jspa?mes...0&tstart=0, (2006).
P. Gralewicz and K. Kowalski, Continuous time evolution from iterated maps and Carleman linearization, http://arxiv.org/abs/math-ph/0002044.
H. S. Wilf, Generatingfunctionology, Academic Press, (1990).
S.C. Woon, Analytic Continuation of Operators -- Operators acting complex s-times, http://arxiv.org/abs/hep-th/9707206.