Dictionary by Dmitrii Kouznetsov. Click here for the
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Ackermann function is a mathematical function A satisfying the recursive equations
A( 1 , z ) = b + z
A( k , 1 ) = b for k>0
A( k , z ) = A( k-1 , A(k,z-1) )
where parameter b is constant; usually, this constant is assumed to be real or even natural. Also, it is assumed that the first argument of the Ackermann function is natural number; the consideration of the holomorphic extension of the Ackermann with respect to the first argument could be a challenging problem.
I do not describe the history of creation of the Ackermann Funcitons, and do not sing PR to Big and Great Wilhelm Ackermann. One may go to some wikipedia if one is interested in the history of science or (due to the specific contingent of the editors) in some fantasies about the history.
The equations above are not sufficient to determine the Ackermann functions for non-integer values of the second argument. The class of these functions can be narrowed with the requirement that function
A is holomorphic with respect to its last argument, at least in the right hand side of the complex plane. It is better to specify the argument ad last, while in some notations the
parameter b is specified as the second argument.
Also, one may require that the function does not have a fast growth in the direction of the imaginary axis: at last, for positive values of the real part of the argument, the growth in the imaginary direction is not faster than polynomial.
Examples
The first functions A, id est, Ak(z = A(k,z) at not so big integer k are well known; one use them without to identify them as Ackermanns.
By the definition, the First Ackermann function is just addition of a constant base b.
Then, using the Third equation of definition, one can express the Second Ackermann A2:
A( 2 , z) = A( 1 , A(2,z-1) ) = b + A(2,z-1)
The equation F(z) = b + F(z-1) has obvious solution F(z) = b z .A( 2 , z) = b z
Using the Second Ackermann, one can express the third one. Repeating the similar substitutions, one finds that the third Ackermann is just exponentiation to base b
A( 3 , z) = bz
In such a way, the Ackermann functions can be built up one by one. The evaluation of the next Ackermann, while the previous one is already implemented, can be done either with the Cauchi integral or with the regular iteration. For the precise evaluation of the multiplication, the good skills in summation are necessary. For the evaluation of exponentiation, one needs both, the summation and multiplication; in addition, the inverse functions, id est, substration and the division also should be implemented.
The fourth Ackermann function, defined in such a way, can be called tetrational, the fifth Ackermann can be called pentational and so on. The first three Ackermanns are entire functions; the fourth one, i.e., the tetrational, has the logarithmic singularity at -2 and cut in the negative direction of the real axis. Behavior of the highest Ackermanns, id est, A(k,z) at k>4, is not yet well investigated, but it is recognized that they show fast growth along the positive direction of the real axis, and this growth with respect to the first argument is faster, that that with respect to the second argument (although the holomorphic extension with respect to the first argument is not yet reported.)
The most of the mathematical analysis of IXX-XX centuries is based on the first three Ackermans: summation, multiplication, exponentiation and the inverse functions. The functions, that can be expressed in terms of these operations, are called elementary functions: log, sqrt, sin, arctan, etc. Some functions can be expressed in terms of simple differential equations with elementary functions; many such functions are called special functions.
Sometimes, one call the solution of some problem exact or analytic, if it is expressed in terms of the special functions or simple integrals with special functions. However, the use of the higher Ackermanns as special functions could extend the class of problems that allow such an exact analytic solution.
Conclusion and references
There are many references related to this topic, and it is difficult to choose the appropriate ones. In addition, I disagree the definitions of the Ackermann I meet in the literature: usually the specific value of b is assumed; often the requirement of the holomorphism with respect to the last argument is omitted. This is reason why I wrote this article.
I do not plan to extend much the text above (except misprints), but I plan to add references if some statements of the article need more examples for the justification.
W.Ackermann. Zum Hilbertschen Aufbau der reelen Zahlen. Mathematische Annalen, v.99, (1928), p.118-133.
J.Ecalle. Theorie des invariants holomorphes. Publications d'mathematiques d'Orsay. no 67-74 09, 1970, Universite Paris XI, U.E.R. Mathematique, 91405 Orsay, France. http://portail.mathdoc.fr/PMO/PDF/E_ECALLE_67_74_09.pdf
R.A.Knoebel. Exponentials reiterated. American Mathematics monthly, v.88, No.4, April 1981, pp, 235-252; especially, see page 247.
D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.
http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
preprint, http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf
D.Kouznetsov. Ackermann functions of complex argument. Preprint ILS, 2008:
http://www.ils.uec.ac.jp/~dima/PAPERS/2008ackermann.pdf
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