Fine-analytic functions in $\mathbb{C}^n$

In this paper we study class of fine-analytic functions in the multidimensional space $\mathbb{C}^n$. The definition of fine-analytic functions in the multidimensional case differs somewhat from the well-known definition of fine-analytic functions on the plane. We give a relationship between classical notion of fine-analyticity and fine-analyticity in $\mathbb{C}^n$.