THE RELATIVELY EXPONENTIAL, LOGARITHMIC AND CIR- CULAR FUNCTIONS IN RECURSIVE FUNCTION THEORY BY R. L. GOODSTEIN of Leicester In this paper we develop a theory of relative integration in recursive function theory and apply it to establish the fundamental properties of the relatively ex- ponential, logarithmic and circular functions which are the 'analogues' in the rational space of recursive function theory of the classical functions of the same names. The present account is self contained and may be read without reference to the literature on recursive function theory. 1 Recursive function theory is a development of a free variable formalisation of arithmetic introduced by Th. Skolem. ~ The elementary formulae of this system are equations between 'terms', and the class of formulae is constructed from the ele- mentary formulae by the operations of the propositional calculus. The terms are the free numeral variables, the sign 0, and the signs for functions. As function signs we have S (x) for the successor function, and signs for functions introduced by re- cursion. The derivation rules comprise the propositional calculus, the substitution of terms for numeral variables, the schema a=b-->(a (a)-->a (b)), the induction schema r (o), a (n)-+~ (S (n)) Q(n) 1 A list of publications on recursive function theory is given in the Bibliography of the author's "Constructive Formalism" (Leicester, 1951). 2 In his paper "Begriindung der elementaren Arithmetik durch die rekurrierende Denkweise ...", Videnskapsselskapets Skri/ter (Kristiania 1923), 2, Vol. I w 7 pp. 3-38. 172 R.L. GOODSTEIN explicit definition, and the recursive schema I (0, a) = ~ (a) / (S (n), a) =fl (n, a, / (y (n), a)) in which /(n, a) is the function introduced by the schema and a (a), ~ (n, a, b) and (n) are functions previously introduced. If y (n)=n, the schema is called primitive recursion; if however ~ (n) is a predecessor of n in a series of natural numbers of transfinite ordinal Q then the recursion is said to be transfinite of ordinal Q.1 Provided that the ordinal does not exceed co~'~, it has been shown that the value, for any assigned argument, of a recursive function of ordinal Q, may be determined by repeated substitution, the number of such substitutions being given by a func- tion of ordinal less than ~. A codification of recursive arithmetic has been given in which all the foregoing axioms and axiom schema are demonstrable3 To complete these introductory remarks it remains only to indicate the passage from the recursive arithmetic of natural numbers to the recursive arithmetic of ia- tional numbers. Following Bernays, we define a rational number to be an ordered triplet (p, q)/r, with r > 0, such that (p, q)/r ~ (p', q')/r' according as pr' 4 q' r ~ p' r + q r'; likewise a recursive function of one or more variables (p, q)/r is a triplet of re- cursive functions of natural numbers (P (p, q, r), Q (p, q, r))/R (p, q, r), with R (p, q, r)> 0, and (writing P for P (p, q, r), P' for P (p', q', r'), etc.) (P', Q')/R'= (P, Q)/R when (p', q')/r'= (p, q)/r. We shall generally denote natural numbers by the letters k, m, n, p, q, r, with or without suffixes, rationals by x and y, and rational functions of a numeral variable n and a rational variable x by /(n, x), g (n, x). The numbers (p, 0)/1, (0, q)/1 are the positive and negative integers respectively and will be denoted as usual by +p, -q, and we take for granted the further details which justify writing (p, q)/r as + (p - q)/r if p > q, - (q - p)/r if p < q and 0 if p=q. i Transfinite recm~ion was introduced by Ackermann in "Zur Widerspruchsfreiheit der Zahlen- theoric", Mathematiache Annalen, Vol. 177 (1940). 2 R. L. GOODSTEIN, "Function theory in an axiom free equation calculus", Proc. London Math. Soc. VoL 48 (1945). RECURSIVE FU~NCTIO~ THEORY 173 The class of rational recursive functions is not increased by applying the original recursion schema to rational functions of a natural number argument. For instance, if we define the rational function /(n, x) by the recursion /(0, x)=0, I(Sn, x)=qa (n, x,/(n, x)) (i) where q~ (n, (p, q)/r, (u, v)/w) ={a(n, p, q, r, u, v, w), b(n, p, ..., u, ...)}/c(n, p, ..., u .... ), then we can find recursive functions un (p, q, r), vn (p, q, r), wn (p, q, r) so that ] (n, (p, q)/r)= (un, v~)/wn; (ii) for if u~, v,, w,~ are defined by the simultaneous recursions u o= v0=0, w o= 1 u~+l (p, q, r)= a (n, u~, v~, w~) Vn+l (p, q, r)=b (n, un, v~, wn) Wn+l(p, q, r)=c(n, Un, Vn, W,) then u~, v~, Wn are recursive and therefore f(n, x) defined by (ii) is recursive and satisfies the rccursion (i). We are now ready to describe the fundamental concepts of recursive function theory. 1. Notation 1.01. If [x] denotes the whole part of x and if for some integer k, [10 kx]=O, we write x=O(k), or x-O(k)=O (so that x=O(k) is equivalent to the recursive relation Ixl < 10-% If there is a recursive function N(ml, m S.... , mp) such that a relation R (n, m> m S.... , rap) holds for all ml, m S.... , mp and all n > N (m D m 2..... mp) then we say that R holds /or majorant n. More generally, if there are recursive functions N o(mJ, m 2.... , rap), N 1 (nl, mp m S.... , rap) such that R (n v n2, ml, m S..... rap) holds for all ml, m., ..... mp and n 1 > N O(m 1, m e..... rap), n 2 --~ ~V1 (nl, ~'1, ms, -", rap) then R 1 (re1_, n2, ml, ..., my) is said to hold for majorant n 1, n 2 (the order of nl, n 2 being material). 174 1~. L. GOODSTEIN 1.1. Equivalence: The recursive functions /(n, x), g(n, x) are said to be equi- valent functions of x, or equal relative to n, if / (n, x) - g (n, x) = 0 (k) for majorant n. Similarly ] (n 1, n2, x), g (n 1, n2, x) are said to be equal relative to nl, n~ if /(n,, n 2, x)- - g (n D nz, x) = 0 (k) for majorant n 1, n 2. 1.2. Recursive convergence. A recursive function /(n, x) of a positive integral variable n and a rational variable x is said to be (recursive) convergent in n, for a~xGb, if there is a recursive N(k,x) l such that N(k+l, x)>N(k,x)>k and for nl>n~>N(k , x), and a<=x~b, / (nl x) / (n~, x) = 0 (k). If N(k, x) may be replaced by a recursive N(k) independent of x then /(n, x) is said to be uniformly rccursive convergent; and if N (k) = k, / (n, x) is in standard/orm. 2 t.3. Relative convergence. /(m, n) is said to be convergent in m, relative to n, if there is a recursive M(k) such that, for ml, m 2_>M(k), ] (ml, n) - ] (m2, n) = 0 (k), for majorant n. 1.4. Relative continuity. /(n, x) is said to be continuous for a<x<b, relative to n, if ](n, x) is recursive convergent for a<x<=b and if there is a strictly in- creasing recursive function c (k) such that, for all xl, x 2 satisfying a<=x 1 <x 2 <= b and x, - x 2 = 0 (c (k)), / (n, xl) - / (n, x2) = 0 (k), for majorant n. 1.5. Relative differentiability. / (n, x) is said to be differentiable for a =< x g b, relative to n, with a relative derivative /1 (n, x), if /(n, x) and /1 (n, x) are recursive convergent for a~x< b and if there is a strictly increasing recursive d (k) such that for all x, x* satisfying a < x < x* <=b and x* - x = 0 (d (k)) we have {/(n, x*) -- ] (n, x)}/(x* - x) -/1 (n, x) = 0 (k), for majorant n. a We observe that, if /(n, x) is relatively differentiable, then both /(n, x) and /1 (n, x) are relatively continuous in (a, b). i When it is rendered necessary by the context we denote the connection of this function N with the function / by attaching / to ~V as a suffix. In the same way the several functions ill (k), c (k) and d (k) of w 1.3, 1.4, 1.5 carry suffixes when needed to avoid ambiguity. 2 Every uniformly convergent function has an equivalent in standard form, for if ] (n, x) is uniformly convergent then ] (N (k), x) is an equivalent in standard form. a Relative differentiability is therefore uniform in x. RECURSIVE ~UNCTION THEORY 175 1.51. Relative continuity and relative differentiability are invariants of the equi- valence relation and any recursive equivalent of a relative derivative of a function /(n, x) is itself a relative derivative of /(n, x). 1 1.6. Recursive differentiability. A recursive /(n, x) is said to be recursively differentiable (for each value of n) with derivative g (n, x), for a g x g b, if there is a recursive d (n, k) such that for all x, x* satisfying a ~< x < x* <~ b and x* - x = 0 (d (n, k)) we have {f (n, x *) - / (n, x)}/(x* - x) - g (n, x) = 0 (k) for each value of n.