Complex Integral 1 Masahiko Yamazaki Shinshu University Nagano , Japan Hiroshi Yamazaki

In this paper t denotes an element of R. The function R2 → C from R× R into C is defined as follows: (Def. 1) For every element p of R × R and for all elements a, b of R such that a = p1 and b = p2 holds (R2 → C)(〈a, b〉) = a+ b · i. Let a, b be real numbers, let x, y be partial functions from R to R, let Z be a subset of R, and let f be a partial function from C to C. The functor ∫ (f, x, y, a, b, Z) yielding a complex number is defined by the condition (Def. 2).