Abstract
Full Text
I. M. Melnik
On Topological Methods in the Theory of Functions of a Complex Variable
(Presented by Academician N. I. Muskhelishvili, 21 XII 1959)
§ 1.
In recent decades a new theory has arisen, relying on topological methods and making it possible to determine the number of critical points of harmonic, pseudoharmonic, analytic, and pseudoanalytic functions from prescribed singularities in a domain and from contour values. A summary exposition of this theory is given in the monograph of M. Morse (¹). For M. Morse, the admissible singularities for harmonic functions are logarithmic poles, and for analytic functions, poles. The simultaneous presence of a polar and a logarithmic singularity at one and the same point is not allowed. Only finite domains are investigated.
In the work of F. D. Gakhov and Yu. M. Krikunov (²), the results of M. Morse’s monograph are generalized and refined in various directions. Some results of the work of F. D. Gakhov and Yu. M. Krikunov have been generalized by T. A. Kolomiytseva (³). As in the monograph (¹), so also in the works (², ³), only single-valued harmonic functions are studied.
In the present note we study the case when a function (f(z)), analytic in a given domain (G) and continuously extendable to the boundary (G), has a finite number of interior and boundary singular points (a_k). In a neighborhood of (a_k), the function (f(z)) has the form
[
f(z)=(z-a_k)^{p_k}\left[g_k(z)\ln^{q_k}(z-a_k)+\psi_k(z)\right]+C,
\tag{1}
]
if (z=a_k) is a finite point;
[
f(z)=z^{p_k}\left[g_k(z)\ln^{q_k}z+\psi_k(z)\right]+C',
\tag{2}
]
if (z=a_k) is the point at infinity; (p_k) and (q_k) are arbitrary integers, (q_k\ne0); the functions (g_k(z)) and (\psi_k(z)) are analytic in a neighborhood of (a_k), if (a_k) is an interior point of the domain (G), and continuous at (a_k) and have first derivatives on the boundary (G) in a neighborhood of (a_k), if (a_k) is a boundary point; (|g_k(a_k)|+|\psi_k(a_k)|\ne0); (C) is a complex constant; (G) is a finite or infinite domain bounded by (\alpha) Jordan curves ((\Gamma_1,\Gamma_2,\ldots,\Gamma_\alpha)=\Gamma). Since an infinite domain (G) can be mapped one-to-one and with preservation of orientation onto a finite domain (D), in the investigation it is sufficient to consider the case of a finite domain (G).
Definition. A point (a_k) in a neighborhood of which (f(z)) has a representation of the form (1) or (2) will be called a power-logarithmic point of the function (f(z)). In the special case when (g_k(z)\equiv0) and (p_k=1), (a_k) is an ordinary point of the function (f(z)). The number (p_k) corresponding to the point (a_k) will be called the order of the power-logarithmic point (a_k).
We shall call a boundary curve (\Gamma_k) exterior if any of its points can be connected with the infinitely distant point by a Jordan curve lying entirely outside (G). If such a connection is impossible, then the boundary curve (\Gamma_k) will be called interior.
In the cases considered here the function (u(x,y)=\operatorname{Re} f(z)) has in (G) a finite number of discontinuity lines leading from the points (a_k) to the exterior boundary curve (\Gamma_\alpha). The discontinuity lines of (u(x,y)) are cuts in (G) that single out single-valued branches of logarithms.
Let (L_k) be a cut in (G) connecting the point (a_k) with the exterior boundary curve (\Gamma_\alpha), and let (t) be the complex coordinate of the points of the line (L_k). By the function (\ln (z-a_k)) we shall mean any of its branches, single-valued in the domain (G) cut along (L_k), and taking on the left side of (L_k) the value (\ln (t-a_k)). If (q_k\ne 1), then we shall assume that the cut (L_k), issuing from the point (a_k=\alpha_k+i\beta_k), passes along the line (t=x+i\beta_k,\ x\le \alpha_k), and that the function (g_k(z)) satisfies the condition (\operatorname{Im} g_k(x+i\beta_k)=0) on (L_k). By elementary calculations one can show that in this case
[
u^+(x,y)-u^-(x,y)=0 \quad \text{on } L_k .
]
If (q_k=1), then, putting in (1)
[
w_k(z)=u_k+iv_k=(z-a_k)^{p_k}g_k(z),
]
we obtain on (L_k)
[
u^+(x,y)-u^-(x,y)=2\pi v_k .
]
In this case we shall consider admissible only those cuts (L_k) on which the function (v_k) is constant, or piecewise constant. It can be shown that, when the function (w_k(z)) is single-valued in (G), there always exists an admissible cut (L_k) connecting (a_k) with the exterior boundary curve (\Gamma_\alpha). If (w_k(z)) is multivalued in (G), then we shall assume that only those cases are considered for which admissible cuts exist.
The admissible cuts (L_k) can always be replaced by admissible cuts (C_k) so that the following conditions are satisfied:
1) The cuts (C_k) do not intersect and have no common points with the interior boundary curves; they do not pass through interior critical points of (u(x,y)) and end on (\Gamma) at ordinary (noncritical) points of the function (u(x,y)).
2) The functions (u^+) and (u^-) (the limiting values of the function (u(x,y)) on (C_k)) have no more than a finite number of points of relative extremum on (C_k); they increase when approaching the point of intersection of (C_k) with the exterior boundary curve (\Gamma_\alpha).
Assume first that on the boundary (\Gamma) there are no power-logarithmic points and that the function (u(x,y)) satisfies the boundary conditions (A) or (C) (((^1)), p. 60) everywhere on (\Gamma), except, possibly, at the points of intersection of the cuts (C_k) with (\Gamma_\alpha). Since the points of intersection of (C_k) with (\Gamma_\alpha) are ordinary points of the function (u(x,y)), the contribution of the boundary index of the function (u(x,y)) from (\Gamma_\alpha) is computed according to the rules set forth in the monograph ((^1)). By the boundary index of the function (u(x,y)) along the contour (\Gamma) relative to (G) we shall mean the sum of the contributions from each curve (\Gamma_k).
Theorem 1. Let (I) be the boundary index of the function (u(x,y)=\operatorname{Re} f(z)) along the contour (\Gamma) relative to (G). Then the equality
[
\sum_{k=1}^{m}(1-p_k)=2-\alpha+I,
\tag{3}
]
holds, where (m) is the number of power-logarithmic points of (f(z)) in (G); (p_k) are the orders of these points; (\alpha) is the number of boundary curves.
In the proof of the theorem the decisive role is played by the following
Lemma. Let (z=a_k) be a power-logarithmic point of (f(z)) of order (p_k), and let (L_k) be an admissible cut issuing from (a_k), on which
(u^+(x,y)-u^-(x,y)=0). There exists a sufficiently small number (r_0) such that the increment (I_{\gamma_k}) of the boundary index of the function (u(x,y)) from the circle (\gamma_k) ((|z-a_k|=r_0)), relative to the domain (|z-a_k|>r_0), is equal to (p_k).
Denote by (G_0) the domain obtained from (G) by removing the closed circular neighborhoods of the points (a_k), not containing other critical points except (a_k), and all cuts on which
[
u^+(x,y)-u^-(x,y)=2\pi c \ne 0;
\tag{4}
]
denote by (\Gamma_0) the boundary of (G_0). It follows from (4) that on the admissible cut (C_k) the functions (u^+(x,y)) and (u^-(x,y)) have the same number of relative extremum points; moreover, the extremum points of (u^+(x,y)) on the left bank coincide with the extremum points of (u^-(x,y)) on the right bank of the cut. Since an extremum entering on the left bank of the cut is exiting on the right, it is easy to show that the increment of the boundary index of (u(x,y)) from (C_k) relative to (G_0) is equal to (-1). With the aid of the lemma and the last conclusion, it is not difficult to obtain a proof of the theorem.
§ 2. Let (a_k=z(s_k)) be an arbitrary point of the contour (\Gamma), ordinary, angular, or a return point, and suppose that in a neighborhood of (a_k) the contour (\Gamma) is smooth to the left and to the right of (a_k), except, perhaps, for the point (a_k). Suppose that
(\Omega(s)=u(s)+iv(s)) is the boundary value of an analytic function (f(z)), having in (\overline G) a finite number of interior and boundary power-logarithmic points (a_k) of orders (p_k), and that (u(s)) between the points (a_k) satisfies boundary conditions (A) or (C). By elementary operations, taking into account the equation of the contour (\Gamma), the function (\Omega(s)) in a neighborhood of (a_k) is transformed to the form
[
\Omega(s)=[z(s)-a_k]^{p_k}{\Phi(s)\ln^{q_k}[z(s)-a_k]+\Psi(s)}+C,
]
where (|\Phi(s_k)|+|\Psi(s_k)|\ne 0).
Around each boundary point (a_k) describe a circle (\gamma_k) ((|z-a_k|=r)) of so small a radius (r) that it intersects (\Gamma) only at two points (b_{1k}) and (b_{2k}), situated at a positive distance from the points of relative extremum of (u(s)) on (\Gamma), so that the function (u(s)) on the arcs (b_{1k}a_k) and (a_kb_{2k}) varies strictly monotonically, and so that the intersection of (\overline G) with the disk (|z-a_k|\le r) contains no other power-logarithmic points of the function (f(z)), except the point (a_k). Denote by (G_0) the domain obtained from (G) by removing the closed neighborhoods of the boundary points (a_k) cut off by the circles (\gamma_k); denote by (\Gamma_0) the boundary of (G_0); denote by (\delta_k) the closed arc of the circle (\gamma_k) entering into (\Gamma_0); (\Gamma_*=\Gamma_0-\sum\delta_k). Let (U(x,y)) be the function defined by the values (\operatorname{Re} f(z)) in (\overline G_0).
By the boundary index of the function (u(s)) along the contour (\Gamma) relative to (G) we shall mean the boundary index of the function (U) along (\Gamma_0) relative to (G_0).
On (\Gamma_*), (U=u(s)). The boundary values of (U) are unknown only on the arcs (\delta_k). However, as the following theorem shows, the increment of the boundary index of the function (U) from the arc (\delta_k) is determined uniquely.
Denote
[
t=(-1)^{q_k}\operatorname{Re}\bigl[(z-a_k)^{p_k}\Phi(s_k)\bigr],
\qquad \text{if } q_k>0 \text{ or } \Psi(s_k)=0;
]
[
t=\operatorname{Re}\bigl[(z-a_k)^{p_k}\Psi(s_k)\bigr],
\qquad \text{if } q_k<0 \text{ or } \Phi(s_k)=0.
]
Theorem 2. Let (I_t) be the increment of the boundary index of the function (t) from the closed arc (\delta_k) relative to (G_0); let (I_{a_k}) be the increment of the boundary index of the function (U) from the closed arc (\delta_k) relative to (G_0). If on the arcs (b_{1k}a_k) and (a_kb_{2k}) the function (t) is strictly monotone, then (I_t=I_{a_k}).
It follows from what has been said that relation (3) is also valid in the case when (f(z)) has a finite number of boundary power-logarithmic points (a_k), if (I) is understood there as the boundary index of the function (u(s)), defined in § 2.
Rostov-on-Don
State University
Received
21 VIII 1959
REFERENCES
- M. Morse, Topological methods in the theory of functions of a complex variable, IL, 1951.
- F. D. Gakhov, Yu. M. Krikunov, Izv. Akad. Nauk SSSR, Ser. Mat., 20, 207 (1956).
- T. A. Kolomiitseva, Izv. Vyssh. Uchebn. Zaved., Mathematics, No. 3 (10), 91 (1959).