On the expressibility of a uniform function of several complex variables as the quotient of two functions of entire character

Abstract

It is a classical fact in the theory of functions of one complex variable that any meromorphic function may be expressed as the quotient of two entire functions without common zeros. When f(x) is a uniform function with essential singularities at finite distance, this theorem may be extended, as was shown by Weierstrasst for a finite number of essential singularities, and by Mittag-Leffler in the general case: f(x) is expressible as the quotient of two functions of entire character (that is, uniform and without poles, but generally both having the same essential singularities as/(x)) without common zeros. Before taking up the corresponding question for several variables, it is convenient to recall the following definitions: The complex variables xy, x2, • • ■, x;l are interior to the region (Sy, S2, • • , Sn) when xy is interior to the region Sy in the xi-plane, ■ • •, xn interior to the region S„ in the «„-plane; the regions Si, ■ • •, S„ may be simply or multiply connected. A uniform function/ ( x\, x2, • • •, xn ) of the complex variables Xi, x%, • • •, xn is meromorphic in (Si, S2, • • ■, Sn) when, in the vicinity of every point ai, a2, • • •, a„ interior to (Si, S2, • • •, Sn), we have