关键词: Complex submanifold, Analytic varieties, Unified formula, Extension, Differentiable function, Integral representation

Abstract: The closure of the bounded domains D in Cnconsists of a chain of the slit spaces,and may be divided into two types. Based on the two types of bounded domains in Cn, firstly using different method and technique we derive the corresponding integral representation formulas of differentiable functions for complex n-m（0 ≤ m < n） dimensional analytic varieties in the two types of the bounded domains. Secondly we obtain the unified integral representation formulas of differentiable functions for complex n-m（0 ≤ m < n） dimensional analytic varieties in the general bounded domains. When functions are holomorphic, the integral formulas in this paper include formulas of Stout[1], Hatziafratis[2] and the author[3],and are the extension of all the integral representations for holomorphic functions in the existing papers to analytic varieties. In particular, when m = 0, firstly we gave the integral representation formulas of differentiable functions for the two types of bounded domains in Cn. Therefore they can make the concretion of Leray-Stokes formula. Secondly we obtain the unified integral representation formulas of differentiable functions for general bounded domains in Cn. So they can make the Leray-Stokes formula generalizations. 

Key words: Complex submanifold, Analytic varieties, Unified formula, Extension, Differentiable function, Integral representation