Reports of the Academy of Sciences of the USSR
1960. Volume 131, No. 2

MATHEMATICS

A. A. TEMLYAKOV

INTEGRAL REPRESENTATIONS

(Presented by Academician M. A. Lavrent’ev on 21 XI 1959)

In preceding papers \((^{1-3})\) I constructed the theory of integral representations for functions analytic in such bicircular domains \(D\) at every boundary point of which the Levi determinant \(L(\Phi)>0\), where the boundary is given by
\[ |z|-\varphi(|w|)\equiv \Phi(w,\overline w,z,\overline z)=0, \]
i.e., for functions regular in bicircular domains whose boundaries are nowhere an analytic hypersurface. In the general case, as is known, \(\varphi(r_1)\) may be constant on an interval beginning with the value \(r_1=0\) (\(r_1\) and \(r_2=\varphi(r_1)\) are conjugate radii of convergence) \((^4)\), and therefore this portion of the boundary of the domain \(D\) may be an analytic hypersurface. Considering the boundary of the domain \(D\) as given in the form \(|w|=\psi(|z|)\), we conclude that a portion of the boundary of the domain \(D\) beginning with the value \(|z|=r_2=0\) may also be an analytic hypersurface. Thus, in the more general case, on the portions \(0\le |w|\le r_1\), \(|z|=R_2\), and \(|w|=R_1\), \(0\le |z|\le r_2\), one may have \(L(\Phi)=0\), and only outside them \(L(\Phi)>0\) (we leave aside the case in which \(L(\Phi)\) vanishes at isolated points of the boundary of the domain \(D\)).

However, the integral representations of both kinds \((^3)\) remain valid for this general case if \(r_1(\tau)\) and \(r_2(\tau)\) are defined in the following way. The function \(r_1(\tau)\), positive and continuously differentiable on the segment \(0\le \tau\le 1\), satisfies the conditions: \(r_1(0)>0\); in the interval \(0<\tau<1\), \(r_1'(\tau)>0\), and \(r_1'(1)=0\). The function \(r_2(\tau)\) is defined in terms of \(r_1(\tau)\) in the same way as before:
\[ r_2(\tau)=R_2\exp\left[-\int_0^\tau \frac{\tau}{1-\tau}\,d\ln r_1'(\tau)\right], \tag{1} \]
where \(R_2\) is a positive constant.

Indeed, since \(r_1'(1)=0\), it may happen that \(r_2(1)>0\) \((^1)\). For example,
\[ r_1(\tau)=2-(1-\tau)^2. \]
Then
\[ r_2(\tau)=\left[2-(1-\tau)^2\right] \left(\frac{\sqrt2+(1-\tau)}{\sqrt2-(1-\tau)}\right)^{1/\sqrt2} (\sqrt2-1)^{\sqrt2}, \]
and we have \(r_1(0)=1\), \(r_1'(1)=0\), \(r_2(1)=2(\sqrt2-1)^{\sqrt2}\). Taking into account that the monomials \(w^m z^n\), \(m\ge 0\), \(n\ge 0\), are invariants with respect to the transformation
\[ \frac{1}{2\pi}\int_0^{2\pi} dt\int_0^1 \frac{d}{du} \left[(u(r_1)u)^m(r_2(\tau)v)^n\right]\,d\tau = w^m z^n \]

for arbitrary continuous functions \(r_1(\tau), r_2(\tau)\) \((^5)\), and \(r_2(\tau)\) is defined in the same way as before, we arrive at the conclusion that in this general case as well, taking into account \(r_1(0)>0,\ r_2(1)>0\), the entire preceding theory of integral representations is preserved.

Thus, the integral representations

\[ F(w,z)=\frac{1}{4\pi^2 i}\int_0^{2\pi}dt\int_0^1 d\tau \int_{|\zeta|=1} \frac{\Phi\left[r_1(\tau)\zeta^n,\ r_2(\tau)\eta^n\right]}{\zeta-u}\,d\zeta, \tag{2} \]

\[ F(w,z)=\frac{1}{4\pi^2 i}\int_0^{2\pi}dt\int_0^1 d\tau \int_{|\zeta|=1} \frac{\zeta F\left[r_1(\tau)\zeta^n,\ r_2(\tau)\eta^n\right]}{(\zeta-u)^2}\,d\zeta, \tag{3} \]

where

\[ u=\tau\left(\frac{w}{r_1(\tau)}\right)^{1/n} +(1-\tau)\left(\frac{z}{r_2(\tau)}\right)^{1/n}e^{it}, \]

\[ \Phi(w,z)=F(w,z)+nwF'_w(w,z)+nzF'_z(w,z), \]

\(n\) is the least integer not smaller than
\[ \sup_{0<\tau<1}\frac{d\ln r_1(\tau)}{d\ln \tau}, \]
hold in the domain
\[ D:\ |w|<r_1(\tau),\ |z|<r_2(\tau),\ 0\le \tau\le 1, \]
i.e., in the domain bounded by three hypersurfaces:
\[ 0\le |w|\le r_1(0),\ |z|=r_2(0); \quad |w|=r_1(\tau),\ |z|=r_2(\tau),\ 0<\tau<1; \quad |w|=r_1(1),\ 0\le |z|\le r_2(1). \]

Here, as we see, there appears a characteristic feature of analytic functions of two complex variables: the values of the function \(F(w,z)\) in the domain \(D\) are determined by its behavior, or by \(\Phi(w,z)\), only on that part of the boundary of the domain \(\overline D\) which is a nonanalytic hypersurface.

Moscow Regional Pedagogical Institute
named after N. K. Krupskaya

Received
12 XI 1959

CITED LITERATURE

\(^{1}\) A. A. Temlyakov, Uch. zap. Moskovsk. obl. ped. inst., 21, 7 (1954).
\(^{2}\) A. A. Temlyakov, Izv. AN SSSR, Ser. Mat., No. 21, 89 (1957).
\(^{3}\) A. A. Temlyakov, DAN, 120, No. 5 (1958).
\(^{4}\) B. A. Fuks, Theory of Analytic Functions of Several Complex Variables, Moscow, 1948.
\(^{5}\) A. A. Temlyakov, DAN, 129, No. 5 (1959).