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|