| 摘 要： | Consider the power series of several complex variables z_1m,…,z_nIn A. in 1962, the domain of the convergence of the series (A) was discussed and the concepts of "domains of the complete n-circular type" (the 2n-dimension) and "conjugate convergent radii" etc. were used. However, the author did not point out how to compute each conjugate radius, only the expression connecting each conjugate convergent radius was given and it is not correct [See《Some Problems of the Theory of Functions of Several Complex Variables (Ⅰ)》, The Convergent Set of the Power Series of Several Complex Variables, Section 13, 《Journal of South China Institutc of Technology》No.1, 1978, PP.82-101] and (Ⅵ) respectively.In 1972, C. pointed out that the convergent set of the series (A) is the same as its absolute convergent set [See 1973, 22, 10 141,1972, 185-195.]. But how large is such a point set? In Problem(I), not only its size had been computed, but also incidentally we obtained the conclusion that the convergent set of the series (A) is the same as its absolute conveagent set CSee remark 6.1].In Problem (Ⅰ), we first computed the convergent set which is also the absolute convergent set of the series of functions of several complex variablesp_0(z_1,…,z_n)+P_1)(z_1,…z_n)+…+P_m(z_1,…,z_n)+…(B)[See section 3], and incidentally we derived Cauchy-Hadamard formula for computing the convergent radius of the power series of a single complex variable. Secondly, we introduced the transformation of models [See section 1] thus the required results were obtained. The concepts of the convergent measure, convergent set and convergent limit etc. were used above.In Problem (Ⅱ)[The Convergent Domain of the Power Series of Several Complex Variables, See 《Journal of South Chiua Institute of Technology》No. 1; 1978, PP. 102- 120], we constructed the convergent domains whieh are also the absolute convergent domains of various dimensions by means of the convergent set which is also the absolute convergent set of the series (A) [See theory 1.8]. The concepts of the convergent radius and convergent domain were used here.In Problem (Ⅲ), the concept of the convergent bound was used, and we pointed out that there is at least one singular manifold and it is at least (n-1) -dimensional on the (2n-1)-dimensional convergent bound, and we gave the equation satisfied by singular manifolds; Two examples were given; in one of which the unique (n-1)—dimensional manifold was computed; and in the other, the only n number of (2n-2) - dimensional manifold were obtained [in the book of, A. cited above, in the discussion on the domain of the complete n-circular type (the 2n-dimension), it was only pointed out that the series (A) has at least one singular point on the bound of the domain, and the author did not point out the way to find the singular point].In Problem (IV), we discussed a wider kind of domains which were named the 2n-dimensional pseudomulticylinder, the 2n-dimensional convergent domain of the series (A) is its special case [See theory 4].A number of the above conclusions also hold for problems of various domains where the dimension number is less than 2n.
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