Full Text
A. A. TEMLYAKOV
INTEGRAL REPRESENTATIONS FOR MEROMORPHIC FUNCTIONS
(Presented by Academician V. I. Smirnov on 23 XI 1961)
1. In papers \((^{1-3})\) I established that if a function \(F(w,z)\) is regular in a bounded complete doubly circular domain \(D \ni (0,0)\) with twice continuously differentiable and analytically convex exterior defining boundary hypersurface \(\Gamma\) (i.e., in the domain \(D\), bounded by the hypersurfaces \(\Gamma_1\{0 \le |w| \le r_1(0),\ |z| = r_2(0)\}\), \(\Gamma\{|w| = r_1(\tau),\ |z| = r_2(\tau),\ 0 < \tau < 1\}\), \(\Gamma_2\{|w| = r_1(1),\ 0 \le |z| \le r_2(1)\}\), where \(r_1(\tau)\) is a positive, continuously differentiable function on the interval \(0 < \tau < 1\), satisfying the conditions \(r_1(0)=\lim_{\tau \to 0} r_1(\tau) \ge 0,\ 0 \le r_1'(\tau) \le \dfrac{r_1(\tau)}{\tau}\)
and
\[ r_2(\tau)=R_2 \exp\left[-\int_0^\tau \frac{\tau}{1-\tau}\, d\ln r_1(\tau)\right], \]
and if \(L[F]=F+wF_w+zF_z\) is a continuous function in the closed domain \(\overline D\), then for \((w_0,z_0)\in D\)
\[ F(w_0,z_0)=\frac{1}{4\pi^2 i}\int_0^{2\pi}dt\int_0^1 d\tau \int_{|\zeta|=1}\frac{L^{(\Gamma)}[F]}{\zeta-u_0}\,d\zeta, \tag{1} \]
where
\[ u_0=\tau\frac{w_0}{r_1(\tau)}+(1-\tau)\frac{z_0}{r_2(\tau)}e^{it}, \]
and \(L^{(\Gamma)}[F]\) is the value of the operator \(L(F)\) on the hypersurface \(\Gamma\{w=r_1(\tau)\zeta,\ z=r_2(\tau)\zeta e^{-it}\}\). Formula (1) expresses \(F(w,z)\) inside the domain \(D\) in terms of the values of the linear differential operator \(L[F]\) on the part \(\Gamma\) of the boundary of the domain \(\overline D\).
In the present note formula (1) is generalized to meromorphic functions \(F(w,z)=\varphi(w,z)/f(w,z)\), where \(\varphi(w,z)\), \(f(w,z)\) are regular functions in the domain \(D\) and \(f(w,z)\not\equiv 0\), and also to functions of a more general nature.
2. First consider functions of the form \(F(w,z)=\varphi(w,z)/f(w/z)\), where \(f(\zeta)\) is an entire function.
Theorem 1. If the product \(f(w/z)F(w,z)\) is a regular function in the domain \(D\) and the operator \(L[f(w/z)F(w,z)]\) is continuous in the closed domain \(\overline D\), then for \((w_0,z_0)\in D\), but \(f(w_0/z_0)\ne 0\),
\[ F(w_0,z_0)=\frac{1}{4\pi^2 i}\int_0^{2\pi}dt\int_0^1d\tau \int_{|\zeta|=1} K(w,z;w_0,z_0)\, \frac{L^{(\Gamma)}[F]}{\zeta-u_0}\,d\zeta, \tag{2} \]
where
\[ K(w,z;w_0,z_0)=f(w/z):f(w_0/z_0),\qquad (w,z)\in\Gamma. \]
Proof. Since the operator
\[ \omega\left[f\left(\frac{\omega}{z}\right)\right]\equiv w f_w\left(\frac{w}{z}\right)+z f_z\left(\frac{w}{z}\right)\equiv 0, \]
it follows that
\[ f\left(\frac{w}{z}\right)L[F] = f\left(\frac{w}{z}\right)F(w,z)+f\left(\frac{w}{z}\right)\omega[F(w,z)] = \]
\[ = f\left(\frac{w}{z}\right)F(w,z) + f\left(\frac{w}{z}\right)\omega[F(w,z)] + F(w,z)\omega\left[f\left(\frac{w}{z}\right)\right] = \]
\[ = f\left(\frac{w}{z}\right)F(w,z) + \omega\left[f\left(\frac{w}{z}\right)F(w,z)\right] = L\left[f\left(\frac{w}{z}\right)F(w,z)\right]. \]
Therefore, taking formula (1) into account,
\[ \frac{1}{4\pi^{2}i}\int_{0}^{2\pi} dt \int_{0}^{1} d\tau \int_{|\zeta|=1}\frac{f(w/z)L^{(\Gamma)}[F]}{\zeta-u_{0}}\,d\zeta = \]
\[ = \frac{1}{4\pi^{2}i}\int_{0}^{2\pi} dt \int_{0}^{1} d\tau \int_{|\zeta|=1}\frac{L^{(\Gamma)}[f(w/z)F(w,z)]}{\zeta-u_{0}}\,d\zeta = f\left(\frac{w_{0}}{z_{0}}\right)F(w_{0},z_{0}), \]
whence the validity of formula (2) follows.
Corollary 1. If \(f(w/z)\equiv 1\), then formula (2) becomes formula (1).
Corollary 2. If
\[ F(w,z)=\frac{\varphi(w,z)}{\prod_{k=1}^{m}(w-\lambda_k z)},\quad \lambda_k\quad (k=1,2,\ldots,m) \]
are constants, then, putting
\[ f(\xi)=\prod_{k=1}^{m}(\xi-\lambda_k), \]
we have
\[ F(w_{0},z_{0})=\frac{1}{4\pi^{2}i}\int_{0}^{2\pi} dt\int_{0}^{1}d\tau \int_{|\zeta|=1}K(w,z;w_{0},z_{0})\frac{L_m^{(\Gamma)}[F]}{\zeta-u_{0}}\,d\zeta, \]
where \(K(w,z;w_{0},z_{0})=z^{m}f(w/z):z_{0}^{m}f(w_{0}/z_{0})\), \(L_m[F]=L[F]+mF(w,z)\).
Remark. Formula (2) expresses the values of the function \(F(w,z)\) inside the domain \(D-P\), where \(P\) is the manifold \(f(w/z)=0\) belonging to \(D\), in terms of the values of the linear differential operator \(L[F]\) on the part \(\Gamma\) of the boundary of the domain \(\overline D\).
- Let now \(f(w,z)\) be an arbitrary function regular in the domain \(D\).
Theorem 2. If the product \(f(w,z)F(w,z)\) is a regular function in the domain \(D\) and the operator \(L[fF]\) is continuous in the closed domain \(\overline D\), then for \((w_{0},z_{0})\in D\), for \(f(w_{0},z_{0})\ne 0\),
\[ F(w_{0},z_{0})= \frac{1}{4\pi^{2}i}\int_{0}^{2\pi} dt\int_{0}^{1}d\tau \int_{|\zeta|=1}K(w,z;w_{0},z_{0}) \frac{L_{\psi}^{(\Gamma)}[F]}{\zeta-u_{0}}\,d\zeta, \tag{3} \]
where
\[ L_{\psi}[F]=L[F]+\psi(w,z)F,\quad \psi(w,z)=\omega(f)/f, \]
\[ K(w,z;w_{0},z_{0})=f(w,z):f(w_{0},z_{0}),\quad (w,z)\in\Gamma. \]
Proof. Since
\[ f(w,z)L_{\psi}[F]=f(w,z)L[F]+f(w,z)\psi(w,z)F(w,z) =f(w,z)L[F]+\omega(f)F=L[fF], \]
then, on the basis of formula (1),
\[ \frac{1}{4\pi^{2}i}\int_{0}^{2\pi} dt \int_{0}^{1} d\tau \int_{|\zeta|=1}\frac{f(w,z)L_{\psi}^{(\Gamma)}[F]}{\zeta-u_{0}}\,d\zeta = \]
\[ = \frac{1}{4\pi^{2}i}\int_{0}^{2\pi} dt \int_{0}^{1} d\tau \int_{|\zeta|=1}\frac{L^{(\Gamma)}[fF]}{\zeta-u_{0}}\,d\zeta = f(w_{0},z_{0})F(w_{0},z_{0}), \]
whence formula (3) follows. With respect to this theorem, the same remark holds as with respect to Theorem 1.
Let \(f_{1}(\xi)\) be an entire function and \(f_{2}(w,z)\) a regular function in the domain \(D\).
A consequence of the two theorems indicated is
Theorem 3. If the product \(f_1(w/z) f_2(w,z) F(w,z)\) is a regular function in the domain \(D\) and the operator \(L_\psi \left[f_1(w/z) f_2(w,z) F(w,z)\right]\) is continuous in the closed domain \(\overline D\), then for \((w_0,z_0)\in D\), but \(f_1(w_0/z_0) f_2(w_0,z_0)\ne 0\),
\[ F(w_0,z_0)=\frac{1}{4\pi^2 i}\int_{0}^{2\pi} dt\int_{0}^{1} d\tau \int_{|\xi|=1} K(w,z;w_0,z_0)\frac{L_\psi^{(\Gamma)}[F]}{\xi-u_0}\,d\xi, \]
where \(L_\psi[F]=L[F]+\psi(w,z)F\), \(\psi(w,z)=\omega[f_2(w,z)]/f_2(w,z)\), \(K(w,z;w_0,z_0)=f_1(w/z)f_2(w,z):f_1(w_0/z_0)f_2(w_0,z_0)\), \((w,z)\in\Gamma\).
- Consideration of \(L_\psi[F]=wF'_w+zF'_z+[1+\psi(w,z)]F\) has led to an integral representation of functions of the form \(F(w,z)=\varphi(w,z)/f_1(w/z)f_2(w,z)\).
Let us now consider an operator of general form
\[ L^*[F]=a(w,z)F'_w+b(w,z)F'_z+F(w,z). \]
It is easy to see that there exist independent functions \(\alpha(w,z)\) and \(\beta(w,z)\) which satisfy the system of equations:
\[ \frac{\alpha\beta'_z-\beta\alpha'_z}{\Delta}=a(w,z),\qquad \frac{\beta\alpha'_w-\alpha\beta'_w}{\Delta}=b(w,z), \]
where \(\Delta=\partial(\alpha,\beta)/\partial(w,z)\). Indeed, these equations are equivalent to the following system of equations:
\[ a(w,z)\frac{\partial\alpha}{\partial w}+b(w,z)\frac{\partial\alpha}{\partial z}=\alpha(w,z), \]
\[ a(w,z)\frac{\partial\beta}{\partial w}+b(w,z)\frac{\partial\beta}{\partial z}=\beta(w,z). \]
Therefore, for the functions \(\alpha(w,z)\) and \(\beta(w,z)\) it is sufficient to take two particular independent solutions of the linear nonhomogeneous first-order partial differential equation
\[ a(w,z)\frac{\partial v}{\partial w}+b(w,z)\frac{\partial v}{\partial z}=v(w,z). \]
This equation also predetermines the class of admissible functions \(a(w,z)\) and \(b(w,z)\). Therefore
\[ L^*[F]=\frac{\alpha\beta'_z-\beta\alpha'_z}{\Delta}F'_w+ \frac{\beta\alpha'_w-\alpha\beta'_w}{\Delta}F'_z+F(w,z). \]
Let the equations
\[ w=\gamma(x,y),\qquad z=\delta(x,y) \]
define a generalized pseudoconformal mapping \({}^{(4)}\) of the domain \(D\) onto some internally branched domain \(D^*\), and let \(x=\alpha(w,z)\), \(y=\beta(w,z)\) be the mapping of the domain \(D^*\) into \(D\). Then
\[ L^*[F]=xF'_x[\gamma(x,y),\delta(x,y)]+yF'_y[\gamma(x,y),\delta(x,y)]+ \]
\[ +F[\gamma(x,y),\delta(x,y)], \]
i.e.,
\[ L^*[F(w,z)]=L\,[F(\gamma(x,y),\delta(x,y))]. \]
Thus, consideration of the operator \(L^*[F(w,z)]\) leads to consideration of the linear differential operator \(L[\Phi(x,y)]\) studied by us, applied to the composite function \(\Phi(x,y)=F[\gamma(x,y),\delta(x,y)]\).
Moscow Regional Pedagogical Institute
named after N. K. Krupskaya
Received
17 XI 1961
CITED LITERATURE
- A. A. Temlyakov, Izv. AN SSSR, Ser. Mat., 21, 89 (1957).
- A. A. Temlyakov, DAN, 120, No. 5 (1958).
- A. A. Temlyakov, DAN, 131, No. 2 (1960).
- B. A. Fuks, Theory of Analytic Functions of Several Complex Variables, Moscow, 1948.