Tetration FAQ Discussion

[Gottfried]

Code:

-- What does half-/ fractional-/continuous-/complex iteration mean?

This problem can only be expressed in terms of the series-paradigm, although
the Reihenalgebra-concept can possibly be seen as an equivalent approach.

The question is - for example -
  given a function f(x) = y
  what is the function g(x) such that g(g(x)) = y

  g(x) is then called the half-iterate of f(x) and is a fractional-iterate
  
The terms half,fractional and continuous are used if the iterator-parameter
is thought as real, but continuous; if the iterator is thought as a general
complex number, sometimes the term continuous is as well used.
  
For real iterator h
  f°h(x) = f°(n+r)(x) = f°n(f°r(x))    where n is integer and r is fractional


Example using powerseries:

For a function f(x), defined by powerseries, with constant term=0 (f(0)=0) and f'(0)=/=0
it is easy to find the half iterate g(x) by manipulation of the powerseries and
equating coefficients at like powers of x:

  Assume f(x) = Ax + Bx^2 + Cx^3 + ...
  target g(x) = ax + bx^2 + cx^3 + ...
  satisfying g(g(x)) = f(x)
  then
  
  g(g(x)) = a g(x) + b g(x)^2 + c g(x)^3 + ...
          =  a*( ax + bx^2 + cx^3 + ...)
           + b*( ax + bx^2 + cx^3 + ...)^2
           + ...
          = a^2 x + (ab + ba^2) x^2 + ...
  = f(x)  =  A  x +    B        x^2 + ...

  then by equating coefficents at like powers of x , either a=+sqrt(A) or a=-sqrt(A)
  and all other coefficients can then uniquely be determined, so we get
  
   g(x) = sqrt(A) x + B/(sqrt(A) + A)*x^2 + ...
  
For general fractional iteration-heights the handling of the appropriate powerseries
is much more complicated and suggests the tools of algebra of infinitely-sized matrices.

Formal analytical handling for general functions is much developed and mostly
based on
  (see:) Abel - functional relation
         Schröder - functional relation
        
  [see : matrix-approach, matrix-logarithm, matrix-diagonalization,
         binomial-expansion using functions, ~ using matrix-operators,
         function-logarithm (ILog) , exponential polynomial interpolation
         <literature>]
  [see : Faa di Bruno-formula, ... ]

  [see further <literature>: iteration-theory, time-series, dynamical systems]
  
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