The Fundamental Theorem of Algebra in Recursive Analysis

Abstract

N is the set of natural numbers. Q is the field of complex rational numbers, i.e. the field of numbers a+bi, where a and b are rational. A sequence σ of elements of Q (i.e. a function σ, σ o:N→Q) is recursive iff there exist recursive functions fj(j=1,2,3,4; fj: N→N) such that for all n in N $$ \sigma (n) = \frac{{{f_1}(n)}} {{{f_2}\left( n \right) + 1}} + \frac{{{f_3}\left( n \right)}}{{{f_4}\left( n \right) + 1}}i$$ Such a sequence σ is recursively convergent iff there exists a recursive function k(k: N→N) such that for all h,j,n in N $$h \geqslant k(n)\Lambda \,j \geqslant k(n) \Rightarrow \left| {\sigma (h) - \sigma (j)} \right| < \frac{1} {{n + 1}}$$ (1)