Abstract: It is well known that the integral with variable upper limit of analytic function is a single value function in the simple connected domain, while the integral with variable upper limit of analytic function in the multiply connected domains is as following: _z0^zf(ζ)dζ, F(z) is not only dependent on the z (z₀ is the fixed point in D), but also depends on the integral path and function f(z) being exact or not in every hole. Therefore F(z) is likely to be multiple valued function. In this paper, we give a new proof method about the integral of analytic function f(z) in the multiply connected domain by the regular covering surface.

Key words: multiply connected domains, regular covering surface, group of covering transformations, exact differential, Abelian group