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Ackermann functions

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  • Аннотация:
    The Matan of IXX-XX century is based mainly on the three Ackermann functions, their inverse functions and their combinations and modifications.

http://budclub.ru/k/kuznecow_d_j/ackermannr.shtml Функции Аскермана (Russian version of this article)
http://budclub.ru/k/kuznecow_d_j/ackermann.shtml Ackermann functions (English version, this text)


The Ackermann function (in slang, "ackermann") is a mathematical function A satisfying the recursive equations

A( 1 , z ) = b + z
A( k , 1 ) = b for k > 1
A( k , z ) = A( k-1 , A(k,z-1) )

where parameter b is constant. Often this constant is assumed to be real or even natural. Usually, it is assumed that the first argument of the ackermann is natural number; the consideration of the holomorphic extension of the ackermann with respect to the first argument is not yet reported. This justifies the notation A( k , z ) = Ak(z)
The sequence {Ak, k=1..} can be considered as set of the Ackermann functions of single variable with single parameter b; this justifies the use of the plural form of the name, id est, the "Ackermann functions".

In this dictionary, the great mathematician Wilhelm Ackermann is not discussed, so, the word "ackermann", (Like "bessel", "airi", or "erfc") refers to the function, not to the person who was first to use and describe it. In order to make difference between Wilhelm Ackermann (1896-1962) and his function, (or functions), in the case of a function, the first latter is not capitalized.

In the notations by Ackermann (1928) and in those by wikipedia (2010), the second argument and the value are displaced for the constant 3: [1,2,3]:
Awiki(n,z)= A(n,z+3)-3.
My co-author calls such a transform "the linear conjugation". The notations with the displaced argument are a little bit more complicated, but, in some sense, equivalent.

The equations above are not sufficient to determine the unique Ackermann functions for non-integer values of the arguments. 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. Also, one may require that the function does not have a fast growth in the direction of the imaginary axis: at least, for positive values of the real part of the argument.

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 Ackermann's.

By the definition, the first ackermann is just addition of a constant base b:

A( 1 , z) = b + z

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 .
In such a way, the second ackermann is just multiplication by constant b.

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 ackermanns can be built up one by one. The evaluation of each ackermann, while the previous one is already implemented, can be done either with the Cauchi integral or with the regular iteration. (Some other methods are also reported, but they seem to be non efficient.) 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, subtraction and the division also should be implemented. The fourth ackermann, defined in such a way, can be called tetrational and denoted with symbol tet; it is shown in the figure for base b=sqrt(2) approximately 1.414..., b=exp(1/e) approximately 1.44..., b=2, b=e approximately 2.71...
The fifth ackermann can be called pentational, the sixth can be called sextational 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, than that with respect to the second argument. The holomorphic extension with respect to the first argument is not yet reported, although I suspect that this is doable.

Conclusion

The most of the mathematical analysis of IXX-XX centuries is based on the first three ackermans: summation, multiplication, exponentiation and their inverse functions. The functions, that can be expressed in terms of these operations in closed form are called elementary functions: log, sqrt, sin, cosh, arctan, etc. Some functions can be expressed in terms of simple differential or integral equations with elementary functions; many such functions are called special functions (for esample, bessel, erfc, fresnelC, fresnelS, etc. ).

Sometimes, one calls the solution of some problem exact or analytic, if it is expressed in terms of the special functions or simple integrals with special functions. The use of the higher ackermanns as special functions could extend the class of problems that allow such an "exact" ("analytic") solution.

References
W.Ackermann. Zum Hilbertschen Aufbau der reelen Zahlen. Mathematische Annalen, v.99, (1928), p.118-133.
http://portail.mathdoc.fr/PMO/PDF/E_ECALLE_67_74_09.pdf 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.
R.A.Knoebel. Exponentials reiterated. American Mathematics monthly, v.88, No.4, April 1981, pp, 235-252; especially, see page 247.
http://www-history.mcs.st-and.ac.uk/history/Biographies/Ackermann.html
http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html 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.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane, preprint
http://www.ils.uec.ac.jp/~dima/PAPERS/2008ackermann.pdf D.Kouznetsov. Ackermann functions of complex argument. Preprint ILS, 2008. (The old definition, with linear conjugated function is used there, and perhaps there are other defects there, but the update is not yet ready)

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Copyleft 2009-2011 by Dmitrii Kouznetsov. This text may be used for free, attribute http://budclub.ru/k/kuznecow_d_j/ackermannr.shtml

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