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    Conformal Mapping
    Conformal Mapping
    Conformal Mapping
    Ebook756 pages7 hours

    Conformal Mapping

    By Zeev Nehari

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    Conformal mapping is a field in which pure and applied mathematics are both involved. This book tries to bridge the gulf that many times divides these two disciplines by combining the theoretical and practical approaches to the subject. It will interest the pure mathematician, engineer, physicist, and applied mathematician.
    The potential theory and complex function theory necessary for a full treatment of conformal mapping are developed in the first four chapters, so the reader needs no other text on complex variables. These chapters cover harmonic functions, analytic functions, the complex integral calculus, and families of analytic functions. Included here are discussions of Green's formula, the Poisson formula, the Cauchy-Riemann equations, Cauchy's theorem, the Laurent series, and the Residue theorem. The final three chapters consider in detail conformal mapping of simply-connected domains, mapping properties of special functions, and conformal mapping of multiply-connected domains. The coverage here includes such topics as the Schwarz lemma, the Riemann mapping theorem, the Schwarz-Christoffel formula, univalent functions, the kernel function, elliptic functions, univalent functions, the kernel function, elliptic functions, the Schwarzian s-functions, canonical domains, and bounded functions. There are many problems and exercises, making the book useful for both self-study and classroom use.
    The author, former professor of mathematics at Carnegie-Mellon University, has designed the book as a semester's introduction to functions of a complex variable followed by a one-year graduate course in conformal mapping. The material is presented simply and clearly, and the only prerequisite is a good working knowledge of advanced calculus.
    LanguageEnglish
    PublisherDover Publications
    Release dateMay 23, 2012
    ISBN9780486145037
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      Conformal Mapping - Zeev Nehari

      Copyright © 1952 by Zeev Nehari.

      All rights reserved.

      This Dover edition, first published in 1975, is an unabridged and unaltered republication of the work originally published by the McGraw-Hill Book Company, Inc. in 1952.

      9780486145037

      Manufactured in the United States by Courier Corporation

      61137X02

      www.doverpublications.com

      PREFACE

      In the preface to the first edition of Courant-Hilbert’s Methoden der mathematischen Physik, R. Courant warned against a trend discernible in modern mathematics in which he saw a menace to the future development of mathematical analysis. He was referring to the tendency of many workers in this field to lose sight of the roots of mathematical analysis in physical and geometric intuition and to concentrate their efforts on the refinement and the extreme generalization of existing concepts. In the intervening years, this trend has become much more pronounced, and it has led to an increasing division of the workers in mathematical analysis into pure and applied mathematicians. One may deplore this development or welcome it—and many, notably among the pure variety, do welcome it—but it seems inevitable that fields like conformal mapping, whose methods are mathematical while their subject matter is derived from physical and geometric intuition, are bound to suffer in such a division.

      The present book tries to bridge the gulf between the theoretical approach of the pure mathematician and the more practical interest of the engineer, physicist, and applied mathematician, who are concerned with the actual construction of conformal maps. Both the theoretical and the practical aspects of the subject are covered, the discussion ranging from the fundamental existence theorems to the various techniques available for the conformal mapping of given geometric figures. It has not been the author’s aim to prove every statement in its utmost generality and under the weakest possible conditions. Wherever such a procedure would have interfered with the clarity of presentation and would have encumbered the proof with a mass of detail likely to obscure the essential ideas, a slightly less general formulation was preferred.

      The book is designed for the reader who has a good working knowledge of advanced calculus. No other previous knowledge is required. The potential theory and complex function theory necessary for a full treatment of conformal mapping are developed in the first four chapters, thus making the reader independent of any other texts on complex variables. These four chapters are also suitable for use as a text for a one-term first course on functions of a complex variable. The remainder of the book may be covered in a one-year graduate course. There is a large number of problems and exercises, which should make the book suitable for both classroom use and self-study. A special chapter is devoted to a detailed discussion of the conformal-mapping properties of a large number of analytic functions which are of importance in the applications.

      The book covers many recent advances in the theory which so far have not yet been incorporated into textbooks, notably in Chap. VII. In keeping with the character of this book as a text, no references to the literature have been given. The interested reader will find detailed bibliographies in S. Bergman’s The Kernel Function and Conformal Mapping (American Mathematical Society, New York, 1950) and in the Appendix by M. Schiffer in R. Courant’s Dirichlet’s Principle (Interscience, New York, 1950).

      The author has tried to present the material as simply and clearly as possible. With this end in view, the proofs of many known results have been simplified either in form or in substance. No claim of completeness is made, and a number of interesting subjects which do not lend themselves to a concise presentation have been omitted.

      ZEEV NEHARI

      Table of Contents

      Title Page

      Copyright Page

      PREFACE

      CHAPTER I - HARMONIC FUNCTIONS

      CHAPTER II - ANALYTIC FUNCTIONS

      CHAPTER III - THE COMPLEX INTEGRAL CALCULUS

      CHAPTER IV - FAMILIES OF ANALYTIC FUNCTIONS

      CHAPTER V - CONFORMAL MAPPING OF SIMPLY-CONNECTED DOMAINS

      CHAPTER VI - MAPPING PROPERTIES OF SPECIAL FUNCTIONS

      CHAPTER VII - CONFORMAL MAPPING OF MULTIPLY-CONNECTED DOMAINS

      INDEX

      CHAPTER I

      HARMONIC FUNCTIONS

      1. Definitions and Preliminary Remarks. In this section we list a number of elementary definitions and results concerning point sets in the plane which will be required in what follows. Any reader who has been exposed to a course in advanced calculus—and it is to such readers that this book is addressed—will be familiar with the concepts and facts in question; they are listed here only for convenience of reference.

      A point in the xy-plane is given by a pair of real numbers (a,b) which are to be interpreted as rectangular coordinates in this plane. A neighborhood-neighborhood—of a point (a,b) is the set of points (x,y) such that (x − a)² + (y − bis a positive number. A point (a,b) is said to be a limit point or accumulation point of a set of points S if every neighborhood of (a,b) contains a point of S distinct from (a,b). This definition clearly implies that every neighborhood of a limit point contains an infinite number of points of S.

      A limit point of S may, or may not, be a point of S. If every limit point of a set belongs to the set, we say that the set is closed. There are two distinct types of limit points of a set, interior points and boundary points. A limit point (a,b) is said to be an interior point of a set S if there exists a neighborhood of (a,b) which consists entirely of points of S. A limit point which is not an interior point of S, that is, a point (a,b) such that any neighborhood of (a,b) contains both points of S and points which do not belong to S, is called a boundary point of S. A set all of whose points are interior points is called an open set. A simple example of an open set is the interior of the unit circle, that is, the set of points (x,y) for which x² + y² < 1. By adding to a set S all its limit points, we obtain the closure of S; obviously, the closure of any set is a closed set. The complement of a set S is the set of all points (x,y) which do not belong to S.

      The set of points (x,y), where x = x(t), y = y(t), and x(t) and y(t) are continuous functions of t in an interval t1 ≤ t ≤ t2, is called a continuous arc. If a continuous arc has no multiple points, that is, if it does not happen that for two different points t′ and t″ in the above interval we have x(t′) = x(t″), y(t′) = y(t″), it is said to be a Jordan arc; intuitively speaking, a Jordan arc is a continuous arc which does not cut itself. A simple example of a Jordan arc is a nonintersecting polygonal line consisting of a finite chain of linear segments. A continuous arc which has exactly one multiple point, namely, a double point corresponding to the end points t1 and t2 of the interval t1 ≤ t ≤ t2, is called a simple closed Jordan curve. For example, the circle x = cos t, y = sin t, 0 ≤ t ≤ 2π, is a simple closed Jordan curve, the double point in question being the point (0,1) which corresponds to both t = 0 and t = 2π.

      A set S of points is said to be connected if any two of its points can be joined by a Jordan arc all of whose points belong to S; here, the Jordan arc may also be replaced by a polygonal arc. An open connected set is called a domain. The Jordan curve theorem states that a simple closed Jordan curve divides the plane into two domains which have the curve as their common boundary. Although the truth of this theorem seems to be evident, its rigorous demonstration is a lengthy and difficult task.¹ One of the two domains into which the plane is divided by a closed Jordan curve is bounded, i.e., there exists a positive constant C such that all the points (x,y) of this domain satisfy x² + y² < C²; this domain is called the interior of the curve. Correspondingly, the exterior of the curve is characterized by the fact that it contains points (x,y) for which x² + y² is arbitrarily large.

      A differentiable, arc is an arc which possesses tangents at all its points; analytically speaking, such an arc is given by the parametric representation x = x(t), y = y(t), t1 ≤ t ≤ t2, where the functions x(t), y(t) have derivatives x′(t), y′(t) which do not both vanish at the same time. If, moreover, these derivatives are continuous—or, to use geometrical language, if the arc possesses a continuously turning tangent—we speak about a smooth arc. A continuous chain of a finite number of smooth arcs will be called a piecewise smooth arc; if this chain of arcs forms a closed nonintersecting curve, we obtain a piecewise smooth curve.

      A crosscut of a domain is a simple Jordan arc which, apart from its end points, lies entirely in the domain; obviously, the end points of the crosscut coincide with boundary points of the domain. A simply-connected domain D is defined by the property that all points in the interior of any simple closed Jordan curve which consists of points of D are also points of D; in particular, the interior of a simple closed Jordan curve is simply-connected. A crosscut connecting two different boundary points of a simply-connected domain D divides D into two simply-connected domains without common points (the points of the crosscut having been removed). If there exist simple closed Jordan curves which consist entirely of points of a domain D and in whose interior there are points not belonging to D, then D is multiply-connected. D is said to be of connectivity n if there exist no more than n − 1 such Jordan curves whose interiors have no points in common and which contain in their interiors points not belonging to D. The circular ring 1 < x² + y² < 4 is clearly of connectivity 2 or, as we shall also say, the circular ring is doubly-connected. To use intuitive language, a bounded domain of connectivity n is a bounded domain with n − 1 holes. By n − 1 appropriately chosen crosscuts, a domain of connectivity n can be transformed into a simply-connected domain. For example, the crosscut 1 < x < 2, y = 0 transforms the circular ring 1 < x² + y² < 4 into a simply-connected domain.

      The restriction to bounded domains made so far can be easily lifted by introducing the point at infinity whose neighborhood is defined as the exterior of a circle of arbitrarily large radius. To the beginner, the assigning of the character of a point to the infinitely distant may seem strange; however, the situation is easily visualized by the well-known stereographic projection of the sphere onto the plane. A sphere of radius unity is placed on the xy-plane in such a fashion that its south pole rests on the point (0,0). To each point P in the plane, a point P′ on the surface of the sphere is made to correspond in the following manner: the north pole is connected with P by a straight line, and P′ is taken to be the point at which this line pierces the surface of the sphere. This clearly establishes a one-to-one correspondence between the points of the xy-plane and the points on the surface of the sphere. The only apparent exception is the north pole of the sphere. Obviously, no finitely distant point in the plane corresponds to this point; however, the outside of a circle x² + y² > M² of sufficiently large radius M corresponds to the inside of an arbitrarily small arctic zone in the neighborhood of the north pole. The correspondence between the plane and the surface of the sphere is therefore made complete if we define the point at infinity, or infinitely distant point, of the plane as the point corresponding to the north pole of the sphere. Statements of the character of those made in this section which involve the point at infinity are to be understood in the sense that they are true with respect to the map of the plane onto the surface of the sphere by means of the stereographic projection. For example, the Jordan curve theorem may now be formulated as follows: A simple closed Jordan curve divides the plane into two simply-connected domains, one of them bounded and the other unbounded. The unbounded domain, i.e., the exterior of the curve, can be characterized by the fact that it contains the point at infinity. An example of a doubly-connected domain containing the point at infinity is given by the set of points of the xy-plane which remain after removal of the points of the circles (x + 2)² + y² ≤ 1 and (x − 2)² + y² ≤ 1. To use intuitive language, a domain of connectivity n containing the point at infinity is obtained by punching n holes out of the plane.

      2. Elementary Properties of Harmonic Functions. A function u(x,y) is said to be harmonic in a domain D if the partial derivatives

      exist and are continuous, and if

      (1)

      at all points of D. U is said to be harmonic at a point P if it is harmonic in a neighborhood of P.

      The partial differential equation (1), which is known as the Laplace equation, and its three-dimensional equivalent

      (2)

      for functions u(x,y,z) of the three variables x, y, z are of fundamental importance in many branches of applied mathematics and mathematical physics. The gravitational, electrostatic, and magnetostatic potentials and many other functions of physical significance are solutions of the equation (2), where x, y, z denote rectangular coordinates in space. If the physical situation described by the equation (2) is such that the function u(x,y,z) does not change in the z-direction, then u is in reality only a function of the two variables x and y, and (2) reduces to (1).

      From the linear character of the equation (1) it is immediately apparent that a linear combination au(x,y) + bv(x,y), where a and b are constants, is a harmonic function if the same is true of the functions u(x,y) and v(x,y) separately. Another elementary consequence of (1) which is easily verified is the fact that the function u(x + a, y + b) is harmonic in a domain obtained by translating D in the x- and y-directions by the amounts −a and −b, respectively, if u(x,y) is harmonic in D.

      An important harmonic function is obtained by asking for those solutions of (1) which depend on the distance from a given point (a,b) only and are independent of the direction in which we proceed from this point. In view of what was said at the end of the last paragraph, we may assume that the point in question is the origin, i.e., the point (0,0). If we introduce polar coordinates r, θ, we are thus asking for solutions of (1) which depend on r only. Transforming (1) to polar coordinates, we obtain

      (1′)

      Since the desired solutions are to depend on r only and not on θ, this reduces to

      The general solution of this differential equation is easily found to be u = A log r + B, where A and B are arbitrary constants. Since a constant is trivially a harmonic function and, as pointed out before, a linear combination of harmonic functions is likewise harmonic, we obtain the result that there is essentially only one harmonic function with the desired properties, namely, the function

      (3)

      This function is harmonic at every finite point of the plane, with the obvious exception of the point (a,b), where the harmonicity of the function breaks down owing to the fact that the required partial derivatives cease to exist. A point of this type is called a singular point, or a singularity of the harmonic function in question.

      Since the harmonic function (3) depends on the arbitrary parameters a and b, we can construct from it other harmonic functions by differentiating it with respect to these parameters. Indeed since

      and

      are harmonic functions, the same is true of the linear combination

      Letting a1 tend to a, we obtain the function

      (4)

      Instead of justifying the passage to the limit we may also confirm directly that (4) is harmonic if (x − a)² + (y − b)² ≠ 0. A similar result is obtained by differentiating log r with respect to b. The point (a,b) again is a singular point of (4), since the required partial derivatives are not defined there. It is worth noting the different character of the singularity of the function (4) at the point (a,b), as compared with that of the function (3) at the same point. The function (3) tends to − ∞ if (x,y) approaches the point (a,b) along any path. The function (4) shows quite a different behavior. If (x,y) approaches (a,b) along the line x = a, the function obviously vanishes for any value of y, and the same is therefore true in the limit y → b. If, on the other hand, (x,y) approaches (a,b) along y = b, the function reduces to (x − a)−1 and the limit for x → a is ∞ or − ∞ , depending on the direction of approach.

      EXERCISES

      1. Show that for any rectilinear nonhorizontal approach to the point (a,b) the function (4) tends to either ∞ or − ∞.

      2. Show that ex sin y, ex cos y, tan−1 [(y − b)/(x − a)] are harmonic functions.

      3. Show that u(xr−2,yr, is a harmonic function, if the same is true of u(x,y). Hint: Introduce polar coordinates and use the form (1′) of the equation (1); observe that the transformation in question consists merely in replacing r by r−1 in the polar form.

      4. If r, θ are polar coordinates and n is an arbitrary real number, show that rn cos nθ and rn sin nθ are harmonic functions.

      5. Show that

      (R, ϕ const.) is a harmonic function, and find its singularity.

      3. Green’s Formula. One of the fundamental formulas of the ordinary calculus is

      The analogous result in two dimensions—known as Gauss’ theorem—is as follows:

      If the functions p(x,y) and q(x,y) are continuous and have continuous first partial derivatives in the closure of a domain D bounded by a piecewise smooth curve Γ, then

      (5)

      where the integral on the right-hand side is a line integral extended over the boundary Γ of D in the positive sense, that is, in such a way that the interior of the domain remains at the left if the boundary is traversed.

      The line integral on the right-hand side of (5) is defined by the ordinary integral

      where

      x = x(t), y = y(t), t1 ≤ t ≤ t2

      , is a parametric representation of the curve Γ. We omit here the proof of Gauss’ theorem since it can be found in any text on advanced calculus.

      Gauss’ theorem remains true if D is a multiply-connected domain. If D is of connectivity n and bounded by n piecewise smooth simple closed curves, it is possible—as pointed out in Sec. 1—to transform D by n − 1 suitably chosen crosscuts into a simply-connected domain D′; these crosscuts may be taken to be piecewise smooth arcs. Since D′ is simply-connected, we may apply to it Gauss’ theorem, where the integrations on both sides of (5) are now to be extended over D′ and its boundary Γ′, respectively. The area integral on the left-hand side of (5) is obviously not affected if D′ is replaced by D. As to the line integral, we observe that while those parts of Γ′ which also belong to Γ are described only once, the crosscuts are described twice, in accordance with the fact that on both edges at the crosscuts there are points of D′. Since r’ is to be described in such a way that the interior of D′ remains at the left, it is clear that the two edges are traversed in different directions. Figure 1 illustrates the situation in the case of a triply-connected domain. For greater clarity, the two edges of the crosscuts have been separated; the sense in which the boundary of D′ is described is indicated by arrows. Since the sign of a line integral is inverted if the direction of integration is changed, it follows that the line integrals over the crosscuts cancel each other. The only surviving line integrals are therefore those extended over the boundary Γ of D. We have thus proved that the identity (5) holds for an arbitrary multiply-connected domain bounded by piecewise smooth curves.

      FIG. 1.

      Consider now a function u(x,y) which has continuous first partial derivatives in D + Γ and a function v(x,y) which has continuous partial derivatives of the first and second order there. The functions

      will then satisfy the hypotheses of Gauss’ theorem. In view of

      it follows from (5) that

      or

      (6)

      The line integral on the right-hand side of (6) can be written in a simpler form if we introduce the differentiation in the direction of the outward pointing normal, denoted by ∂/∂n. By the rules of partial differentiation

      where (x,n) and (y,n) denote the angles between the outward pointing normal and the positive x- and y-axis, respectively. Since the direction cosines of the tangent are dx/ds and dy/ds, where s , it follows that

      Hence

      Using this identity and the abbreviation vxx + vyy = Δv, we can finally bring (6) into the form

      (7)

      This identity, which is of fundamental importance in the theory of harmonic functions, is known as Green’s formula; the names Green’s identity and Green’s theorem are also used. Sometimes, (7) is referred to as Green’s first formula, to distinguish it from another formula—Green’s second formula—which is an immediate consequence of (7) If both u(x,y) and v(x,y) are supposed to possess continuous first and second derivatives, the roles of these two functions in (7) may be interchanged. Doing so, we obtain the companion formula

      If we subtract this formula from (7), we arrive at Green’s second formula

      (8)

      Both (7) and (8) are generally referred to as Green’s formula.

      4. Applications of Green’s Formula. As a first application, we identify the function v(x,y) in (7) with a function harmonic in D + Γ, and set u(x,y) ≡ 1. Since, in view of (1), Δv = 0 in D + Γ, it follows from (7) that

      (9)

      An easy consequence of this identity is the mean value theorem: If v(x,y) is harmonic in a closed circle, the value of υ at the center of the circle is the arithmetic mean of its values on the circumference of the circle. Analytically expressed,

      (10)

      where v(x,y) is harmonic in the circle (x − a)² + (y − b)² ≤ r².

      To prove this theorem, we apply (9) to the case in which Γ is the circumference (x − a)² + (y − b)² = ρ², 0 ≤ ρ ≤ rand ds = ρ dθ, we have

      or

      which shows that the integral is independent of ρ. Its value for ρ = 0 is 2πv(a,b); if we set this equal to its value for ρ = r, the identity (10) follows.

      Another result which can be easily deduced from (9) is the maximum principle: If the function v(x,y) is harmonic in a domain D, it cannot attain its absolute maximum or minimum at an interior point of D unless v(x,y) reduces to a constant.

      Proof: Suppose that v(x,y) does attain its maximum M at some interior point or points of D, and denote by S the set of all points of D at which v(x,y) = M. If v(x,y) is not constant, S cannot contain all points of D. Accordingly, there must exist a boundary point of S, say P, which is an interior point of D. If r, θ denote polar coordinates with reference to the point P, for arbitrary values of θ and sufficiently small r, since v is supposed to attain its absolute maximum at P. Integrating over this small circle, we obtain

      According to (9), however, this integral is equal to zero. Clearly, this is only possible if ∂v/∂r = 0 on the whole circumference of the small circle in question and hence also on any smaller circle. It follows that v(x,y) is constant in a small circle surrounding P, which contradicts our assumption that P is a boundary point of S. This proves the theorem for the case of the maximum. The reader will have no difficulty in modifying the argument slightly in order to prove the corresponding result for the minimum.

      The above statement of the maximum principle is of a negative character; it denies the possibility of a harmonic function attaining its maximum at an interior point. It is in the nature of things that nothing more can be said in the case in which the harmonic function is defined only in an open set. If, however, it is known in addition that v is continuous in the closure of D, we can obtain more precise information. From the well-known result that a function which is continuous in a closed set attains its maximum at a point of the set it follows then that v must attain its maximum either in D or on its boundary. Since we have proved that the first alternative is excluded, it follows that the maximum must be attained at a point of the boundary. We have thus proved the following corollary of the maximum principle:

      If v(x,y) is harmonic in a domain D and continuous in the closure of D, then both the maximum and minimum values of v in the closure of D are attained on the boundary.

      The maximum principle leads to an important uniqueness property of harmonic functions. Suppose that u(x,y) and v(x,y) are harmonic in a domain D and continuous in the closure of D and that u = v on the boundary of D; the latter hypothesis is also expressed by saying that u and v have the same boundary values. Consider now the harmonic function w = u − v which, by hypothesis, vanishes on the boundary of D. By the maximum principle, w attains both its maximum and its minimum on the boundary. Since it vanishes there identically, both the maximum and the minimum of w in D are zero; hence, w is identically zero throughout D and the harmonic functions u and v are identical. In other words, a harmonic function is completely determined by its boundary values.

      EXERCISES

      1. Multiplying (10) by r and integrating, with respect to r, from 0 to R, show that the mean-value theorem remains true if the arithmetic mean of the values of v on the circumference of the circle (x − a)² + (y − b)² = R² is replaced by the arithmetic mean of the values of v in the circle (x − a)² + (y − b)² ≤ R².

      2. If the function v is harmonic in a domain D and on its boundary Γ, show by means of Green’s formula that

      and that equality will hold only if v reduces to a constant.

      3. Use the result of the preceding exercise in order to show that (a) a harmonic function with vanishing boundary values is identically zero and that (b) a harmonic function with vanishing normal derivatives at all points of the boundary reduces to a constant.

      4. Let u(x,y) be a nonnegative solution of the partial differential equation

      Δu =p(x,y)u,

      where p(x,y) is continuous and p(x,y) > 0 in a domain D. Using the ordinary necessary conditions for the existence of a local maximum, show that u(x,y) cannot attain its maximum in D in the interior of this domain.

      6. Show that the function u of the preceding exercise satisfies the inequality Δ(u²) ≥ 0 in D.

      5. The Green’s Function and the Boundary Value Problem of the First Kind. Let D be a smoothly bounded domain and Γ its boundary, and let the function w = w(x,y) be harmonic in D and have continuous first partial derivatives in D denotes the distance of (x,y) from a point (ξ,η) in D, then log r is harmonic in D + Γ except at the point (ξ,η), and the same is therefore true of the function

      (11)

      The result we are about to derive depends on the application of Green’s formula (8) to a function of the type (11). Since h(x,y) has a singularity at the point (ξ,η), it is clearly not permissible to take the entire domain D as the domain of integration. We circumvent this difficulty by deleting from D a small circle of center (ξ,η) and radius ∈. In the remaining domain, which we shall denote by D , h(x,y) is harmonic.

      We shall now apply the special form of (8) which is obtained if both u and v are harmonic functions in D; since, in this case, Δu = 0, Δv = 0, the left-hand side of (8) vanishes. The resulting formula is of such frequent occurrence as to warrant formulation as a separate result:

      If u and v are harmonic in a smoothly bounded domain D and have continuous first derivatives in the closure D + Γ of D, then

      (12)

      We now identify v with the function h defined in (11) and take u to be an arbitrary function which is harmonic in D and has continuous first derivatives in D + Γ. The domain D, to which we apply (12) is bounded by Γ and by the circumference Cand center (ξ,η). Since this circle is outside D, and the boundary of D, is to be traversed in the positive sense—that is, leaving D, at the left—C . is clearly described in the clockwise direction. It thus follows from (12) that

      (13)

      where the circle C is now to be described counterclockwise. Introducing polar coordinates in the integral over C, and using (11), we obtain

      The first integral is, by the mean-value theorem, equal to 2πu(ξ,η). The other two integrals are bounded in the neighborhood of (ξ,ηtend to zero, we therefore obtain

      . Comparing the last formula with (13), we thus have arrived at the following result:

      If h is of the form (11), where w is harmonic in D, and u is likewise harmonic in D and both u and w have continuous first derivatives in the closure D + Γ of D, then

      (14)

      where (ξ,η) is any point of D. This result is sometimes referred to as Green’s third formula.

      A striking application of the identity (14) is obtained if h(x,y) is identified with the Green’s function g(x,y;ξ,η) of D. This function, which is of fundamental importance both in the theory of harmonic functions and in the theory of conformal mapping, is defined as follows:

      The Green’s function g(x,y;ξ,η) of a domain D with respect to a point (ξ,η) in D is of the form

      (15)

      where g1 is harmonic in D; if (x,y) tends to any point of the boundary of D, g tends to zero.

      The proof for the existence of the Green’s function of an arbitrary domain will be postponed to a later chapter (Sec. 4, Chap. V); it will then be obtained as a by-product of an existence theorem in the theory of conformal mapping. We shall meanwhile proceed on the assumption that our domain D does have a Green’s function and that—except at the point (ξ,η)—this function has continuous first derivatives in the closure of the smoothly bounded domain D.

      The Green’s function g is zero on the boundary Γ of D. Identifying g with the function h in (11), we thus conclude from (14) that

      Before giving a formal statement of this remarkable result, we introduce a notation which will permit us to formulate much of the following work in a more compact fashion. A point (x,y) will be denoted by the symbol z which stands for the complex number z = x + iy; similarly, the point (ξ,η) will be denoted by ζ = ξ + iη. At this stage, this so-called complex notation is purely formal and its use does not require familiarity with the properties of complex numbers or variables, The symbol g(x,y;ξ,η) will thus be replaced by g(z,ζ), but it should be perfectly clear that this does not imply that g is a function of the variables z and ζ. We shall also occasionally employ z, ζ, etc., as subscripts—such as in dsz, ∂/∂nz—in order to emphasize the variables to which these differentials or differentiations refer. With these notations the above result reads as follows:

      Let u(z) be harmonic in the closure D + Γ of a smoothly bounded domain D, and let g(z,ζ) denote the Green’s function of D with respect to a point ζ of D. If u(z) is the boundary value of u at a point z ∈ Γ, then

      (16)

      Formula (16) makes it possible to compute a harmonic function if its boundary values—and, of course, the Green’s function of the domain in question—are known. A question which presents itself naturally in this connection is the following: If, subject to certain regularity conditions, an arbitrary function U(z) is given on the boundary Γ, does there always exist a harmonic function u(ζ) in D whose boundary values coincide with U(z)? We shall show at a later stage that in the case in which the boundary function U(z) is piecewise continuous on Γ the answer is in the affirmative. The problem of constructing a harmonic function with a given set of boundary values is known as the boundary value problem of the first kind, or the Dirichlet problem. If the Green’s function of a domain is known, the corresponding Dirichlet problem is solved by (16). Everything depends therefore on the explicit knowledge of the Green’s function for the domain under discussion. In the case of a general domain D, the determination of the Green’s function can be a matter of considerable difficulty.

      It is worth noting that the construction of the Green’s function is equivalent to the solution of a particular boundary value problem. Indeed, since g(z,ζ) vanishes for z ∈ Γ, the boundary values of the function g1(z,ζ) in(15) coincide with those of log r, where r is the distance between ζ and z. Hence, if we construct a function u(z,ζ) with the boundary values log r, then the function − log r + u(z,ζ) vanishes on the boundary. Since it further has the prescribed singularity at the point ζ, it necessarily is identical with the Green’s function.

      We now give the proofs of two important properties of the Green’s function. The first of these is:

      The Green’s function g(z,ζ) of a domain D is positive throughout D.

      By (15), g(z,ζ) is of the form − log r + g1(z,ζ), where g1(z,ζ) is harmonic throughout D and r denotes the distance between the points z and ζ. Near the point ζ, g1(z,ζ) is bounded while − log r > 0) is taken small enough, g(z,ζ) will therefore be positive on the circumference C and center ζ. Consider now the values of g(z,ζ) in the domain D obtained from D and center ζ. g(z,ζ) is harmonic in D. The total boundary of D consists of the boundary Γ of D and the circumference C . On Γ, g(z,ζ) = 0; on C , g(z,ζ) > 0. By the maximum principle, g(z,ζ) cannot attain its minimum inside D . Hence, g(z,ζ) is positive throughout D ; since ε can be made arbitrarily small, the above statement is proved.

      Next, we derive the so-called symmetry property of the Green’s function: The Green’s function g(z,ζ) of a domain D is symmetric with respect to the two points z and ζ; that is,

      (17)

      Let ζ and t be two given points of D. The Green’s functions g(z,ζ) and g(z,t) are harmonic in D with the exception of the points ζ and t, respectively. If we delete from D two small circles C (ζ) and C,(t) of radius ε and centers ζ and t, respectively, both functions will therefore be harmonic in the domain D, denotes the boundary of D —consisting of the boundary r of D and the two circles C (ζ) and C (t)—it follows from the identity (12) that

      In this integration, the boundary Γ is described in the positive sense, while the circumferences C (ζ) and C (t) are described in the negative sense. The integral over Γ vanishes, since both g(z,ζ) and g(z,t) vanish there. With the abbreviations g(z,ζ) = g1, g(z,t) = g2, we have therefore

      g1 has a logarithmic singularity of the type (11) at the point ζ while g2 is harmonic in the interior of C (ζ). By (14), the value of the integral over C (ζ) is therefore 2πg2(ζ). Similarly, the value of the integral over C (t), as obtained from (14), is −2πg1(t), the negative sign arising from the fact that g1 and g2 appear in the same order in both integrals although the roles of ζ and t are interchanged. Hence, g2(ζ) = g1(t) or, i . view of the definition of the functions g1 and g2, g(ζ,t) = g(t,ζ). This proves (17).

      We end this section with the explicit determination of the Green’s function in the case in which D is the circle CR of radius R with center at the origin. If r, θ are polar coordinates in the (x,y) plane and ρ,ϕ the polar coordinates of a point ζ inside CR(ρ < R), we shall show that the Green’s function of CR is of the form

      (18)

      Let r1 denote the distance between the points z and ζ, and denote by r2 the distance between z and the point inverse to ζ with respect to the circumference CR, that is, the point with the polar coordinates R²/ρ, ϕ. By elementary trigonometry, we have

      Comparison with (18) shows that the function g(z,ζ) defined by (18) can be written in the form

      (18′)

      It was shown in Sec. 2 that the function log r, where r denotes the distance of the variable point z from a fixed point a, is a harmonic function for z ≠ a. Hence, log r1 and log r2, and therefore also the function g(z,ζ) of (18′), are harmonic functions. The possible singularities of g(z,ζ) are the points from which the distances r1 and

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