Full Text
UDC 517.544
MATHEMATICS
M. Ya. KURGANSKAYA, Yu. I. CHERSKII
THE CONJUGATION PROBLEM FOR THREE ANALYTIC FUNCTIONS
(Presented by Academician N. I. Muskhelishvili on May 4, 1970)
Let \(L\) be a simple smooth closed contour dividing the complex plane into an interior domain \(S^{+}\), containing the point \(z=0\), and an exterior domain \(S^{-}\), containing the point at infinity. Let us construct one more exterior domain \(E^{-}\) as follows. Join the points \(z=0\) and \(z=\infty\) by a simple cut intersecting \(L\) at the single point \(t_{0}\). In what follows, by \(\sqrt z\) we shall mean the fixed branch having a discontinuity along the indicated cut. The locus of the points \(\sqrt t\) and \((-\sqrt t)\), where \(t\) runs over \(L\), forms a simple smooth closed contour \(\Gamma\). As \(E^{-}\) we take the domain lying outside this contour. It is obvious that if \(z \in E^{-}\), then also \((-z) \in E^{-}\).
Problem. Find functions \(\Phi^{+}(z)\) and \(\Psi^{+}(z)\), analytic in \(S^{+}\) and continuous in \(S^{+}+L\), and also a function \(\Phi^{-}(z)\), analytic in \(E^{-}\), continuous in \(E^{-}+\Gamma\), and equal to zero at infinity, under the boundary conditions
\[ \begin{aligned} \Phi^{+}(t) &= A(t)\Phi^{-}(\sqrt t)+G(t),\\ \Psi^{+}(t) &= B(t)\Phi^{-}(-\sqrt t)+H(t), \end{aligned} \qquad t \in L, \tag{1} \]
where \(A(t), B(t), G(t)\), and \(H(t)\) are functions given on \(L\) satisfying a Hölder condition; \(A(t)\) and \(B(t)\) do not vanish anywhere on \(L\).
Below it is shown that problem (1), in its nature and in the method of solution, is related to the well-known conjugation problem (the Riemann problem) \((^{1,2})\).
- Let us first consider the “jump” problem
\[ \varphi^{+}(t)=\varphi^{-}(\sqrt t)+g(t), \qquad \psi^{+}(t)=\varphi^{-}(-\sqrt t)+h(t), \]
\[ t\in L, \qquad \varphi^{-}(\infty)=0. \tag{2} \]
Introduce the new unknown functions
\[ \varphi_1^{-}(z)=\varphi^{-}(\sqrt z)-\varphi^{-}(-\sqrt z), \qquad \varphi_2^{-}(z)=1/\sqrt z\,[\varphi^{-}(\sqrt z)-\psi^{-}(-\sqrt z)], \]
\[ z\in S^{-}. \]
These functions are analytic in \(S^{-}\), continuous in \(S^{-}+L\), and equal to zero at infinity. Setting further \(\varphi_1^{+}=\varphi^{+}+\psi^{+}\), \(\varphi_2^{+}=\varphi^{+}-\psi^{+}\), from conditions (2) we arrive at two independent conjugation problems
\[ \varphi_1^{+}(t)=\varphi_1^{-}(t)+g(t)+h(t), \qquad t\in L; \tag{3} \]
\[ \varphi_2^{+}(t)=\sqrt t\,\varphi_2^{-}(t)+g(t)-h(t), \qquad t\in L. \tag{4} \]
Solving problems (3) and (4) by the known method and using the results of N. I. Muskhelishvili on the behavior of a Cauchy-type integral with density having a discontinuity (see, in particular, \((^{1})\), p. 77), we arrive at the theorem.
Theorem 1. Let the given function \(g(t)+h(t)\) satisfy the Hölder condition everywhere on \(L\), and let the given function \(g(t)-h(t)\) satisfy
satisfies this condition on \(L\) everywhere except, possibly, the point \(t_0\), where a discontinuity of the first kind is allowed for this function.
Then problem (2) has, in the class of functions analytic in the above-indicated domains and continuous up to the boundary, a unique solution \((L\) is traversed counterclockwise):
\[ \varphi^{+}(z)=\frac{1}{4\pi i}\int_L \frac{g(\tau)+h(\tau)}{\tau-z}\,d\tau+ \frac{\sqrt{z-t_0}}{4\pi i}\int_L \frac{g(\tau)-h(\tau)}{\sqrt{\tau-t_0}(\tau-z)}\,d\tau,\qquad z\in S^{+}; \tag{5} \]
\[ \psi^{+}(z)=-\frac{1}{4\pi i}\int_L \frac{g(\tau)+h(\tau)}{\tau-z}\,d\tau- \frac{\sqrt{z-t_0}}{4\pi i}\int_L \frac{g(\tau)-h(\tau)}{\sqrt{\tau-t_0}(\tau-z)}\,d\tau,\qquad z\in S^{+}; \tag{6} \]
\[ \varphi^{-}(z)=\frac{1}{4\pi i}\int_L \frac{g(\tau)+h(\tau)}{\tau-z^2}\,d\tau+ \frac{z\sqrt{z^2-t_0}}{4\pi i\sqrt{z^2}}\int_L \frac{g(\tau)-h(\tau)}{\sqrt{\tau-t_0}(\tau-z^2)}\,d\tau,\qquad z\in E^{-}. \tag{7} \]
The limiting values \(\varphi^{+}(t)\), \(\psi^{+}(t)\), and \(\varphi^{-}(t)\) satisfy the Hölder condition. \(\sqrt{w-t_0}\) denotes the branch of the root having a discontinuity along a part of the cut indicated earlier, connecting \(t_0\) with infinity.
2. We proceed to the problem of factorization of the coefficients \(A(t)\) and \(B(t)\). By analogy with (5)—(7), we write formulas in which the logarithms are single-valued functions, having, possibly, discontinuities at the single point \(t_0\):
\[ X_0^{+}(z)=\exp\left\{\frac{1}{4\pi i}\int_L \frac{\ln A(\tau)B(\tau)}{\tau-z}\,d\tau+ \frac{\sqrt{z-t_0}}{4\pi i}\int_L \ln\frac{A(\tau)}{B(\tau)}\,d\tau/\sqrt{\tau-t_0}\times (\tau-z)\right\},\qquad z\in S^{+}; \tag{8} \]
\[ \Omega_0^{+}(z)=\exp\left\{\frac{1}{4\pi i}\int_L \frac{\ln A(\tau)B(\tau)}{\tau-z}\,d\tau- \frac{\sqrt{z-t_0}}{4\pi i}\int_L \ln\frac{A(\tau)}{B(\tau)}\,d\tau/\sqrt{\tau-t_0}\times (\tau-z)\right\},\qquad z\in S^{+}; \tag{9} \]
\[ X_0^{-}(z)=\exp\left\{\frac{1}{4\pi i}\int_L \frac{\ln A(\tau)B(\tau)}{\tau-z}\,d\tau+ \frac{z\sqrt{z^2-t_0}}{4\pi i\sqrt{z^2}}\int_L \ln\frac{A(\tau)}{B(\tau)}\,d\tau/\sqrt{\tau-t_0}\times (\tau-z^2)\right\},\qquad z\in E^{-}. \tag{10} \]
The properties of the functions \(X_0^{+}\), \(\Omega_0^{+}\), and \(X_0^{-}\) depend on the index
\[ \varkappa=\operatorname{Ind} A(t)+\operatorname{Ind} B(t) =\frac{1}{2\pi i}\,[\ln A(\tau)B(\tau)]_{L}. \tag{11} \]
Let the function \((z-t_0)^{\varkappa/2}\) be analytic in \(S^{+}\), and let the function
\(\omega(z)=(z^2-t_0)^{\varkappa/2}\) be analytic in \(E^{-}\), with the possible exception of the point \(z=\infty\), and satisfy the condition
\(\omega(1/\bar t)=(t-t_0)^{\varkappa/2}\) for \(t\in L\). On the basis of formulas of N. I. Muskhelishvili ((1), § 22) we obtain
Theorem 2. The functions
\[ X^{+}(z)=(z-t_0)^{\varkappa/2}X_0^{+}(z),\qquad \Omega^{+}(z)=(-1)^{\varkappa}(z-t_0)^{-\varkappa/2}\Omega_0^{+}(z) \tag{12} \]
are analytic in \(S^{+}\) and continuous in \(S^{+}+L\); the limiting values \(X^{+}(t)\) and \(\Omega^{+}(t)\) satisfy the Hölder condition on \(L\).
The function
\[ X^{-}(z)=(z^2-t_0)^{-\varkappa/2}X_0^{-}(z) \tag{13} \]
is analytic in \(E^{-}\) (for \(\varkappa<0\) it has a pole of order \(|\varkappa|\) at infinity), and is continuously extendable to the contour \(\Gamma\); the limiting value \(X^{-}(t)\) satisfies the Hölder condition on \(\Gamma\).
The formulas are valid
\[ X^+(t)=A(t)X^-(\sqrt{\bar t}),\qquad \Omega^+(t)=B(t)X^-(-\sqrt{\bar t}),\qquad t\in L. \tag{14} \]
The functions (12) and (13) play the role of the canonical solution of the conjugation problem. These functions do not vanish anywhere at a finite distance.
3. Let us proceed to the direct solution of problem (1). Using the factorization (14), we represent condition (1) in the form
\[ \Phi^+(t)/X^+(t)=\Phi^-(\sqrt{\bar t})/X^-(\sqrt{\bar t})+G(t)/X^+(t), \]
\[ \Psi^+(t)/\Omega^+(t)=\Phi^-(-\sqrt{\bar t})/X^-(-\sqrt{\bar t})+H(t)/\Omega^+(t),\qquad t\in L. \tag{15} \]
Consider the unknown function \(\Phi^-(z)/X^-(z)\). On the basis of the properties of the function \(X^-(z)\) and of the condition \(\Phi^-(\infty)=0\) adopted by us, we conclude that the function \(\Phi^-(z)/X^-(z)\) for \(\varkappa\le 0\) has at infinity a zero of order not less than \(|\varkappa|+1\), while for \(\varkappa>0\) this function may have a pole of order not greater than \(\varkappa-1\). Therefore the indicated function can be represented in the form
\[ \Phi^-(z)/X^-(z)=\varphi^-(z)+P_{\varkappa-1}(z), \tag{16} \]
where \(\varphi^-(z)\) vanishes at infinity, and \(P_{\varkappa-1}(z)\) is a polynomial of degree not exceeding \(\varkappa-1\). For \(\varkappa\le 0\) we put \(P_{\varkappa-1}\equiv 0\).
Introducing also the notation
\[ \Phi^+(z)/X^+(z)=\varphi^+(z),\qquad \Psi^+(z)/\Omega^+(z)=\psi^+(z), \tag{17} \]
we give conditions (15) the form of the jump problem (2), where now
\[ g(t)=G(t)/X^+(t)+P_{\varkappa-1}(\sqrt{\bar t}),\qquad h(t)=H(t)/\Omega^+(t)+P_{\varkappa-1}(-\sqrt{\bar t}). \tag{18} \]
Using Theorem 1, we arrive at the following results.
Theorem 3. If the index \(\varkappa\) defined by formula (11) is positive, then problem (1) is unconditionally solvable, and the homogeneous problem has exactly \(\varkappa\) linearly independent solutions. The general solution can be constructed in quadratures by formulas (16), (17), (5)—(7), (18), (12), (13), and (8)—(10), where \(P_{\varkappa-1}(z)\) is a polynomial of degree \(\varkappa-1\) with arbitrary complex coefficients.
Theorem 4. In the case \(\varkappa=0\), problem (1) is unconditionally solvable and has a unique solution. This solution can be constructed in quadratures by the formulas indicated in Theorem 3, where \(P_{\varkappa-1}\equiv 0\).
As an example, let us write out the function \(\Phi^-(z)\). Using equality (16), where \(P_{\varkappa-1}\equiv 0\), we obtain
\[ \Phi^-(z)=X^-(z)\left\{ \frac{1}{4\pi i}\int_L \left(\frac{G(\tau)}{X^+(\tau)}+\frac{H(\tau)}{\Omega^+(\tau)}\right) \frac{d\tau}{\tau-z^2} +\right. \]
\[ \left. +\frac{z\sqrt{\bar z^2-t_0}}{4\pi i\sqrt{\bar z^2}} \int_L \left(\frac{G(\tau)}{X^+(\tau)}-\frac{H(\tau)}{\Omega^+(\tau)}\right) \frac{d\tau}{\sqrt{\tau-t_0}(\tau-z^2)} \right\},\qquad z\in E^-. \tag{19} \]
We shall also arrive at an expression of the form (19) when solving problem (1) in the case \(\varkappa<0\). However, in the latter case the function \(X^-(z)\) has at infinity a pole of order \(|\varkappa|\), as a result of which problem (1) will be solvable not for all functions \(G(t)\) and \(H(t)\) satisfying the Hölder condition. In the usual way (cf. \((^2)\), pp. 118—119) we derive the necessary and sufficient solvability conditions:
\[ \int_L \bigl(G(\tau)/X^+(\tau)+H(\tau)/\Omega^+(\tau)\bigr)\tau^{k-1}\,d\tau=0, \qquad k=1,2,\ldots,-[(\varkappa+1)/2]. \tag{20} \]
\[ \int_L \left(\frac{G(\tau)}{X^+(\tau)}-\frac{H(\tau)}{\Omega^+(\tau)}\right) \left\{ \left. \frac{d^{k-1}}{dz^{k-1}} \frac{\sqrt{1-t_0z}}{1-\tau z} \right\}_{z=0} \right\} \frac{d\tau}{\sqrt{\tau-t_0}}=0, \]
\[ k=1,2,\ldots,[(-\varkappa+1)/2]. \tag{21} \]
Here the symbol \([x]\) denotes the greatest integer not exceeding \(x\).
Theorem 5. In the case \(\varkappa < 0\), for the solvability of problem (1) the conditions (20) and (21), whose number is equal to \(|\varkappa|\), are necessary and sufficient. If these conditions are fulfilled, then the problem has a unique solution, which can be obtained from the formulas indicated in Theorem 3, where \(P_{\varkappa-1} \equiv 0\).
Let us note that the limiting values \(\Phi^{+}(t)\), \(\Psi^{+}(t)\), and \(\Phi^{-}(t)\) of the functions obtained as a result of solving problem (1) satisfy the Hölder condition and, moreover, \(\Phi^{-}(\overline{\gamma t_{0}})=\Phi^{-}(-\overline{\gamma t_{0}})\).
- The problem considered is a particular case of the following problem of conjugation of analytic functions, whose number is equal to \(n+1\). Suppose that on different copies of the complex plane there are given \((n+1)\) domains \(S_1^{+},\ldots,S_n^{+}\) and \(E^{-}\), bounded by simple smooth closed contours \(L_1,\ldots,L_n\) and \(\Gamma\). Any of the indicated domains may contain the infinitely distant point. On the contours a positive direction of traversal is specified so that the corresponding domains remain on the left. We divide \(\Gamma\) into the sum of \(n\) simple open arcs \(\gamma_k\), \(k=1,\ldots,n\), and on the contours \(L_k\) we specify one point \(t_k\), \(k=1,\ldots,n\). Let \(\alpha_k(t)\) be functions defined on \(L_k\), one-to-one (with the exception of \(t_k\)) and continuously mapping \(L_k\) onto \(\gamma_k\) with a change of the direction of traversal, \(k=1,\ldots,n\). The points \(t_k\) correspond to the endpoints of the arcs \(\gamma_k\). It is required to find functions \(\Phi_1^{+}(z),\ldots,\Phi_n^{+}(z)\), analytic in the domains \(S_1^{+},\ldots,S_n^{+}\), respectively, continuous in \(S_k^{+}+L_k\), and also a function \(\Phi^{-}(z)\), analytic in \(E^{-}\), continuous in \(E^{-}+\Gamma\), from \(n\) boundary conditions
\[ \Phi_k^{+}(t)=A_k(t)\Phi^{-}[\alpha_k(t)]+G_k(t), \qquad t\in L_k,\ k=1,\ldots,n. \tag{22} \]
Here \(A_k(t)\) and \(G_k(t)\) are given functions on \(L_k\), satisfying the Hölder condition, \(k=1,\ldots,n\). It is also assumed that there exist derivatives \(\alpha_k'(t)\) satisfying the Hölder condition and having no zeros.
In the case when all the domains \(S_k^{+}\) coincide, and the functions \(\alpha_k(t)\) are formed from different branches of the function \(z^{1/n}\), problem (22), like problem (1), admits a solution in quadratures.
Odessa State University
named after I. I. Mechnikov
Received
17 IV 1970
CITED LITERATURE
- N. I. Muskhelishvili, Singular Integral Equations, Moscow, 1968.
- F. D. Gakhov, Boundary Value Problems, Moscow, 1963.