Functional sequences and Taylor series with a fuzzy complex number as an argument

Abstract

This article considers functional sequences ƒ_n(A) with fuzzy complex number A for an argument. The convergences lim_n→∞ƒ'_n(z)=ƒ(z) and lim_n→∞ƒ'_n(x)=ƒ'(x) are assumed to be uniform inside each circle supp A. Due to analyticity, the conditions of point-wise convergence of derivatives and finiteness of the number of solutions for equation ƒ(z)=w with respect to z for each w inside each circle supp A are satisfied. The paper proposes the sufficient conditions for the convergence ƒ_n(A) in the sense that the sequence of membership functions μ_ƒn(A)(w) converges point-wise. The convergence lim_n→∞μ_ƒn(A)(w)=μ_ƒ(A)(w) is proved for all points w∈X, except such w=ƒ(z), that z is a discontinuity point of μ_A(z), or ƒ'(z)=0. As a particular case of a sequence ƒ_n(A), the generalization of Taylor series ƒ(z)=∑_i=0^∞ƒ⁽ⁱ⁾(z₀)/i!(z-z₀)ⁱ is considered for an analytical function ƒ(z) for the case of fuzzy complex argument z=A. The convergence of the series is considered in the sense of point-wise convergence of the partial sum μ_Sn(A)(w), where S_n(z)=∑_i=0^∞ƒ⁽ⁱ⁾(z₀)/i!(z-z₀)ⁱ.