scholarly journals On the Boundary Value of a bounded analytic Function of several complex Variables

1945 ◽  
Vol 21 (3-10) ◽  
pp. 308-312
Author(s):  
Masatsugu TSUJI
Keyword(s):  
Analytic Function ◽  
Boundary Value ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Bounded Analytic Function

Linear forms in algebraic points of Abelian functions. II

1976 ◽  
Vol 79 (1) ◽  
pp. 55-70 ◽  
Author(s):  
D. W. Masser
Keyword(s):  
Analytic Function ◽  
Maximum Modulus ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Maximum Modulus Principle ◽  
Linear Forms ◽  
Extrapolation Procedure ◽  
Abelian Functions

In this paper we continue to develop the apparatus needed for the proof of the theorem announced in (11). We retain the notation of (11) together with the assumptions made there about the field of Abelian functions. This section deals with properties of more general functions holomorphic on Cn. When n = 1 the extrapolation procedure in problems of transcendence is essentially the maximum modulus principle together with the act of dividing out zeros of an analytic function. For n > 1, however, this approach is not possible, and some mild theory of several complex variables is required. This was first used in the context of transcendence by Bombieri and Lang in (2) and (12), and we now give a brief account of the basic constructions of their papers.

Maximum modulus algebras and singularity sets

1980 ◽  
Vol 86 (3-4) ◽  
pp. 327-331 ◽  
Author(s):  
John Wermer
Keyword(s):  
Analytic Function ◽  
Maximum Principle ◽  
Maximum Modulus ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Ideal Space ◽  
Analytic Structure ◽  
Analogous Condition ◽  
Classical Theorem ◽  
Analytic Variety

SynopsisA classical theorem of Hartogs gives conditions on the singularity set of an analytic function of several complex variables in order for such a set to be an analytic variety. A result of E. Bishop from 1963 gives an analogous condition of the maximal ideal space of a uniform algebra in order for this space to have analytic structure. We show that algebras of functions satisfying a maximum principle serve to explain both of these results.

scholarly journals Functional Dependence and Analytic Functions

1979 ◽  
Vol 22 (3) ◽  
pp. 367-369
Author(s):  
P. Ramankutty
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Function Theory ◽  
Functional Dependence ◽  
Complex Variables ◽  
Several Complex Variables ◽  
The Real ◽  
Cauchy Theorem ◽  
Complex Valued ◽  
Constant Complex

AbstractWithout appealing to the Cauchy theorem or its corollaries, it is proved that the real and imaginary parts of a non-constant complex-valued analytic function of several complex variables are functionally independent. This unifies and generalizes some results sporadically treated in standard treatises on function theory.

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