The Schwarz Function and Its Applications

The Schwarz Function and Its Applications

PHILIP J. DAVIS
Volume: 17
Copyright Date: 1974
Edition: 1
Pages: 241
https://www.jstor.org/stable/10.4169/j.ctt5hh99x
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    The Schwarz Function and Its Applications
    Book Description:

    H. A. Schwarz showed us how to extend the notion of reflection in straight lines and circles to reflection in an arbitrary analytic arc. Notable applications were made to the symmetry principle and to problems of analytic continuation. Reflection, in the hands of Schwarz, is an antianalytic mapping. By taking its complex conjugate, we arrive at an analytic function that we have called here the Schwarz Function of the analytic arc. This function is worthy of study in its own right and this essay presents such a study. In dealing with certain familiar topics, the use of the Schwarz Function lends a point of view, a clarity and elegance, and a degree of generality which might otherwise be missing. It opens up a line of inquiry which has yielded numerous interesting things in complex variables; it illuminates some functional equations and a variety of iterations which interest the numerical analyst. The perceptive reader will certainly find here some old wine in relabelled bottles. But one of the principles of mathematical growth is that the relabelling process often suggests a new generation of problems. Means become ends; the medium rapidly becomes the message. This book is not wholly self-contained. Readers will find that they should be familiar with the elementary portions of linear algebra and of the theory of functions of a complex variable.

    eISBN: 978-1-61444-017-8
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. Table of Contents
    (pp. ix-x)
  3. ACKNOWLEDGEMENTS
    (pp. xi-xii)
    PHILIP J. DAVIS
  4. CHAPTER 1 PROLOGUE
    (pp. 1-4)

    In the year 1968–1969, Professor Mary Cartwright was a visiting member of the Division of Applied Mathematics at Brown University. This was a year of turmoil at Brown—particularly curricular turmoil—and in the course of one of our division meetings Miss Cartwright remarked that when she was a student all mathematics majors were required to know a proof of the nine-point circle theorem. Since the nine-point circle now has a distinct flavor of beautiful irrelevance, Miss Cartwright seemed to be telling us that we should not be too dogmatic as to what constitutes a proper mathematics curriculum. Fashion...

  5. CHAPTER 2 CONJUGATE COORDINATES IN THE PLANE
    (pp. 5-6)

    Plane Euclidean geometry and, in particular, many of the topics which normally appear in advanced synthetic or inversive geometry may be expeditiously carried out by working with complex coordinates. The real plane is converted into the complex plane by assigning the complex number$z = x + iy$,$i = \sqrt { - 1} $to the real point (x, y). If we wish to recoverxandyfromz,it is convenient to introduce the conjugate${\bar z}$quantity by means of the equations

    (2.1)$z = x + iy$,$\bar z = x - iy$,

    and hence the inverse transformation is given by

    (2.2)$x = \frac{1}{2}\left( {z + \bar z} \right)$,$y = \frac{1}{{2i}}\left( {z - \bar z} \right)$,

    The non-independent qunatitiesz,${\bar z}$are called the conjugate coordinates of...

  6. CHAPTER 3 ELEMENTARY GEOMETRIC FACTS
    (pp. 7-12)

    Given two distinct points${z_1}$and${z_2}$in the complex plane, the equation

    (3.1)$z = t{z_1} + (1 - t){z_2}$,treal,

    describes all the points on the straight line joining${z_1}$and${z_2}$. The pointzdivides the line segment from${z_1}$to${z_2}$in the ratio

    (3.2)$r = (1 - t)/t$

    The distance from${z_1}$to${z_2}$is$\rho = |{z_1} - {z_2}|$or

    (3.3)$\rho = \sqrt {({z_1} - {z_2})({{\bar z}_1} - {{\bar z}_2})} $

    The determinant

    (3.4)$D = \left( {\begin{array}{*{20}{c}} z & {\bar z} & 1 \\ {{z_1}} & {{{\bar z}_1}} & 1 \\ {{z_2}} & {{{\bar z}_2}} & 1 \\ \end{array}} \right)$

    vanishes for$z = {z_1}$and for$z = {z_2}$and sinceDis linear inxandy, the equation

    (3.5)$D = 0$

    must be the equation in conjugate coordinates of the straight line through the points${z_1}$and${z_2}$. Upon expansion, (3.5) can...

  7. CHAPTER 4 THE NINE-POINT CIRCLE
    (pp. 13-18)

    Given a triangleT,the three midpoints of the three sides, the three feet of the altitudes from each vertex to the opposite side, and the three midpoints between the orthocenter (i.e., the intersection of the altitudes) and the vertices all lie on a circle. The radius of this circle is half that of the circle circumscribingT.The center of the nine-point circle lies half way between the circumcenter and the orthocenter. This is thetheorem of the nine-point circle(see Fig. 4.1).

    In proving this theorem it is advantageous to select the circumcircle as the unit circle$z\bar z = 1$...

  8. CHAPTER 5 THE SCHWARZ FUNCTION FOR AN ANALYTIC ARC
    (pp. 19-28)

    From straight lines and circles, we turn to analytic arcs and curves. We suppose that the arcCis written in rectangular form

    (5.1)$f\left( {x,y} \right) = 0$.

    In conjugate coordinates, this becomes

    (5.2 )$f\left( {\frac{{z + \bar z}}{2},\frac{{z - \bar z}}{{2i}}} \right) \equiv g\left( {z,\bar z} \right) = 0$.

    Example. Consider the real conic written in the matrix form

    (5.3)$X'AX = 1$,

    where

    $X = \left( \begin{array}{l} x \\ y \\ \end{array} \right)$,$A = \left( {\begin{array}{*{20}{c}} a & b \\ b & c \\ \end{array}} \right)$,

    and the prime designates the transpose. If$Z = \left( {\frac{z}{z}} \right)$, then$z = MX$with

    $M = \left( {\begin{array}{*{20}{c}} 1 & i \\ 1 & { - i} \\ \end{array}} \right)$.

    From (2.4),$X = \frac{1}{2}M*Z$,$X' = \frac{1}{2}Z'\bar M$so that,

    (5.4)$Z'\left( {\bar MAM*} \right)Z = 4$

    is the equation of the conic in conjugate coordinates.

    If

    (5.5)$\delta = \det (\bar MAM*)$

    then

    (5.6)$\delta = - 4\det A$.

    Since the conic is an ellipse or a hyperbola according as the discriminant$\delta > 0$or$\delta < 0$, the...

  9. CHAPTER 6 GEOMETRICAL INTERPRETATION OF THE SCHWARZ FUNCTION; SCHWARZIAN REFLECTION
    (pp. 29-40)

    Reflection in a straight line. Let${z_1}$and${z_2}$be two distinct points. The transformation

    $T(z) = \frac{{|{z_1} - {z_2}|}}{{({z_1} - {z_2})}}\left( {z - {z_2}} \right)$

    is a rigid motion which brings${z_2}$to the origin and${z_1}$thex-axis. Its inverse is${T^{ - 1}}(z) = (({z_1} - {z_2})/|{z_1} - {z_2}|)z + {z_2}$. The transformation$R(z) = \bar z$is a reflection in thexaxis. Hence the composite function${T^{ - 1}}RT$reflectszin the lineldetermined by${z_1}$and${z_2}$. It is easily computed to be

    (6.1)$z* = {T^{ - 1}}RT(z) = \frac{{{z_1} - {z_2}}}{{{{\bar z}_1} - {{\bar z}_2}}}(\bar z - {{\bar z}_2}) + {z_2}$.

    Comparing this with (5.11) or (3.7) we see that

    (6.1')$z* = \overline {S(z)} $,

    where$S(z)$is the Schwarz Function ofl.(See Fig. 6.1.)

    Reflection or inversion in a circleC:$|z-{z_0}|=r$.

    The definition...

  10. CHAPTER 7 THE SCHWARZ FUNCTION AND DIFFERENTIAL GEOMETRY
    (pp. 41-48)

    In this chapter, we shall relate the derivatives of the Schwarz Function of an analytic arcCto the slope, curvature, etc., of the arc. Let the point$z = r{e^{i\theta }}$lie onC. Then

    (7.1)${r^2} = z\bar z = zS(z) = |S(z){|^2}$alongC.

    Since$\bar z/z = {e^{ - 2i\theta }}$, we have

    (7.2)$\theta = \frac{i}{2}\log \left( {\frac{{\bar z}}{z}} \right) = \frac{i}{2}\log \left( {\frac{{S(z)}}{z}} \right)$, alongC.

    AlongCwe have,

    (7.3)$\frac{{d\bar z}}{{dz}} = \frac{{dx - idy}}{{dx + idy}} = \frac{{1 - iy'}}{{1 + iy'}}$, where$y' = \frac{{dy}}{{dx}}$.

    However, alongC,$\bar z = S(z)$, and since the derivative an analytic function is independent of the direction in which increments are taken, we have

    (7.4)$S'(z) = \frac{{d\bar z}}{{dz}} = \frac{{1 - iy'}}{{1 + iy'}}$.

    Since$dx - idy = \overline {dx + idy} $, it follows from (7.3) and

    (7.4) that

    (7.5)$|S'(z)| = 1$alongC.

    The equality (7.5) may also be obtained as a consequence...

  11. CHAPTER 8 CONFORMAL MAPS, REFLECTIONS, AND THEIR ALGEBRA
    (pp. 49-88)

    In Chapter 6, we defined an analytic arcCas the image of the real segment$a \le t \le b$under a one-to-one conformal mapf. We found that if a point$z = f(t)$is in a neighborhood ofC, the reflection* ofzinCis defined by$z* = f(\bar t)$· Since$S(z) = \overline {z*} $, we have$S(z) = \overline {f(\bar t)} = \bar f(t)$and since$t = {f^{ - 1}}(z)$, we have$S(z) = \bar f({f^{ - 1}}(z))$. We can write this as

    (8.1)$S = \bar f{f^{ - 1}}$,

    from which we obtain

    (8.1')$S' = \bar f'{f^{ - 1}}/f'{f^{ - 1}}$.

    If the parametertin [0, 1] is changed by means of

    (8.2)$t = g(t')$,$0 \le t' \le 1$,

    wheregis real analytic on the real${t'}$axis:$\begin{array}{l} g = \bar g \\ z* = f(\bar t \\ \end{array}$, then we can compute...

  12. CHAPTER 9 WHAT FIGURE IS THE $\sqrt -1$ POWER OF A CIRCLE?
    (pp. 89-102)

    9.1 The$\sqrt { - 1} $power of a circle and spirals. Of course one can make definitions to suit one’s own fancy; we shall naturally answer the question in the title of this Chapter in terms of the Schwarz Function.

    Suppose thatCis an analytic arc (see Fig. 9.1), and let$z \ne 0$be a variable point on it. Draw the half ray from 0 tozcutting the arc and making an angle$\psi = \psi (z)$with the arc. The angle$\psi $will be an analytic function ofz. The clinant of the half ray is$\bar z/z$while that of the arc is...

  13. CHAPTER 10 PROPERTIES IN THE LARGE OF THE SCHWARZ FUNCTION
    (pp. 103-108)

    Given an analytic arc, the global properties of its Schwarz Function$S(z)$are of considerable interest. The first observation to make, derived from an inspection of the Schwarz Functions we have displayed explicitly in Chapter 5, is that with the exception of the straight line, all the Schwarz Functions have singularities. This is true, as we shall now prove. We base our proof upon the identity (6.11) backed up by a theorem of Pólya on the composition of entire functions.

    Suppose that$f(z), g(z), h(z)$are entire functions related by

    (10.1)$f(z) = g(h(z))$.

    Suppose further that$h(0) = 0$. Let$F(r)$,$G(r)$,$H(r)$donate respectively...

  14. CHAPTER 11 DERIVATIVES AND INTEGRALS
    (pp. 109-134)

    A function ofxandy,$f(x,y)$can be written as a function ofz,$\overline z $through the identity

    (11.1)$F(z,\overline z ) = f\left( {\frac{{z + \overline z }} {2},\frac{{z - \overline z }} {{2i}}} \right)$

    Since

    $\frac{{\partial F}} {{\partial z}} = \frac{{\partial F}} {{\partial x}}\frac{{\partial x}} {{\partial z}} + \frac{{\partial F}} {{\partial y}}\frac{{\partial y}} {{\partial z}} = \frac{{\partial F}} {{\partial x}}\left( {\frac{1} {2}} \right) + \frac{{\partial F}} {{\partial y}}\left( {\frac{1} {{2i}}} \right)$,

    we have

    (11.2)$\frac{{\partial F}} {{\partial z}} = \frac{1} {2}\left( {\frac{{\partial F}} {{\partial x}} - i\frac{{\partial F}} {{\partial y}}} \right)$

    Similarly,

    (11.2’)$\frac{{\partial F}} {{\partial \overline z }} = \frac{1} {2}\left( {\frac{{\partial F}} {{\partial x}} + i\frac{{\partial F}} {{\partial y}}} \right)$

    This suggests that it will be convenient to introduce the operators

    (11.3)$\frac{\partial } {{\partial z}} = \frac{1} {2}\left( {\frac{\partial } {{\partial x}} - i\frac{\partial }{{\partial y}}} \right)$,$\frac{\partial } {{\partial \overline z }} = \frac{1} {2}\left( {\frac{\partial } {{\partial x}} + i\frac{\partial } {{\partial y}}} \right)$.

    Inversely,

    (11.4)$\frac{\partial } {{\partial x}} = \frac{\partial } {{\partial z}} + \frac{\partial } {{\partial \overline z }}$,$\frac{\partial } {{\partial y}} = i\left( {\frac{\partial } {{\partial z}} - \frac{\partial } {{\partial \overline z }}} \right)$.

    The directional derivative of$f(x,y)$in the direction a is defined by

    ${D_\alpha }f(x,y) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h\cos \alpha ,y + h\sin \alpha )}} {h}$

    $ = \frac{{\partial f}} {{\partial x}}\cos \alpha + \frac{{\partial f}} {{\partial y}}\sin \alpha $

    Using (11.4), one obtains

    (11.5)${D_\alpha }F(z,\overline z ) = {F_z}{e^{i\alpha }} + {F_{\overline z }}{e^{ - i\alpha }}$

    To obtain directional derivatives along and normal to an analytic arcCwith Schwarz Function$S(z),$we proceed as follows. Let${z_0} \in C$and let$\psi $be the angle the tangent toC...

  15. CHAPTER 12 APPLICATION TO ELEMENTARY FLUID MECHANICS
    (pp. 135-148)

    The theory of analytic functions is often applied to plane problems of fluid flow, heat flow, electrostatics, and elasticity. We select fluid flow as a prototype, and show its relationship to the Schwarz Function.

    LetDbe a region in the complex z-plane. Avelocity fieldor asteady flowonDcomprises two functions

    (12.1)$u(z) = u(x,y)$;$v(z) = v(x,y)$

    which provide thexand theyvelocity components of a fluid particle located at$z = x + iy$. One way of visualizing the flow is by erecting the vector${\text{v}} = u{\text{i}} + v{\text{j}}$at each point Z₀. Introduce thecomplex velocityof the flow by means of...

  16. CHAPTER 13 THE SCHWARZ FUNCTION AND THE DIRICHLET PROBLEM
    (pp. 149-152)

    By theDirichlet ProblemorFirst Boundary Value Problemin two dimensions is meant the following: given a regionBin the$(x,y)$plane and given a function$u(z*)$defined for the boundary pointsz* ofB, find a function$u(z)$which is harmonic inB(i.e., satisfies$\Delta u = 0$at interior points ofB) and for which${\lim _{z \leftarrow z*}}u(z) = u(z*)$in some sense. For the purposes of this essay, we limit ourselves to a regionBwhose boundary is an analytic curve with Schwarz Function$S(z)$and where the boundary function$u(z*)$is continuous.

    The solution of the Dirichlet problem is well...

  17. CHAPTER 14 SCHWARZ FUNCTIONS OF SPECIFIED TYPE
    (pp. 153-172)

    In view of the fact that certain constructions require that$S(z)$be meromorphic inside the curve, it is of considerable interest to characterize analytic curves with this property. We shall obtain a characterization in terms of the mapping function of the region bounded by the curve.

    LetBbe a region of the complex plane. The class$aC( = aC(B))$will designate the functionsfwhich are regular inBand continuous in$B + \partial B$.

    A linear functionalL,defined on functions regular inB,is said to be of class$D = D(B)$if it can be expressed in the form

    (14.1)$L(f) = \sum\limits_{n - 1}^N {\sum\limits_{k = 0}^{nk} {{a_{nk}}} } {f^{(k)}}({Z_n})$

    where...

  18. CHAPTER 15 Schwarz Functions and Iterations
    (pp. 173-206)

    The object of iteration theory is to study the properties of the set of successive transformations${\tau ^n}(p),n = 0,1,{\text{ }}{\text{. }}{\text{. }}{\text{. }}$of the pointP.Among the many questions of interest are:

    (a) What are the invariant subsets ofτ? What are the fixed points ofτ? What are the periodic points (cycles) ofτ? i.e., what are the points${P_1},{P_2},{\text{ }}{\text{. }}{\text{. }}{\text{. }},{P_k}$such that$\tau ({p_i}) = {P_{i + 1}}$where the subscripts are taken modκ? What are the invariant curves?

    (b) What are the convergence properties of$\tau$? What are the limit sets?

    (c) What are theergodicproperties ofτ? i.e., what can be said about the...

  19. CHAPTER 16 DICTIONARY OF FUNCTIONAL RELATIONSHIPS
    (pp. 207-208)
  20. CHAPTER 17 BIBLIOGRAPHICAL AND SUPPLEMENTARY NOTES
    (pp. 209-218)
  21. 18. BIBLIOGRAPHY
    (pp. 219-224)
  22. INDEX
    (pp. 225-228)