Conformal Mapping on Riemann Surfaces

by Harvey Cohn

Overview

The subject matter loosely called "Riemann surface theory" has been the starting point for the development of topology, functional analysis, modern algebra, and any one of a dozen recent branches of mathematics; it is one of the most valuable bodies of knowledge within mathematics for a student to learn.
Professor Cohn's lucid and insightful book presents an ideal coverage of the subject in five parts. Part I is a review of complex analysis analytic behavior, the Riemann sphere, geometric constructions, and presents (as a review) a microcosm of the course. The Riemann manifold is introduced in Part II and is examined in terms of intuitive physical and topological technique in Part III. In Part IV the author shows how to define real functions on manifolds analogously with the algebraic and analytic points of view outlined here. The exposition returns in Part V to the use of a single complex variable z. As the text is richly endowed with problem material — 344 exercises — the book is perfect for self-study as well as classroom use.
Harvey Cohn is well-known in the mathematics profession for his pedagogically superior texts, and the present book will be of great interest not only to pure and applied mathematicians, but also engineers and physicists. Dr. Cohn is currently Distinguished Professor of Mathematics at the City University of New York Graduate Center.

Product Details

ISBN-13:	9780486153292
Publisher:	Dover Publications
Publication date:	04/07/2014
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	352
File size:	23 MB
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About the Author

A Distinguished Professor of Mathematics at the City University of New York Graduate Center, Harvey Cohn is well known for his pedagogically superior texts.

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Conformal Mapping on Riemann Surfaces

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By Harvey Cohn

Dover Publications, Inc.

Copyright © 1967 Harvey Cohn
All rights reserved.
ISBN: 978-0-486-64025-9

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CHAPTER 1

ANALYTIC BEHAVIOR

At first we restrict our interest to functions which are well behaved where denned. All well-behaved functions are "equally" well behaved under this restriction, for assuming differentiability in a neighborhood of a point, we find a valid power series to exist.

DIFFERENTIATION AND INTEGRATION

1-1 Analyticity

Let us start with the concept of a real function u(x) defined in an interval I: (a < x < b). There is a certain interest in "degrees of regularity" of the function u(x) as represented by the smoothness of the curve or its tangent, etc. Thus we consider restrictions of u(x) to classes C, C', C", ..., C(∞), A as follows:

C: u(x) is continuous for all x in I.

C': u'(x) exists and is continuous for all x in I.

C": u?(x) exists and is continuous for all x in I.

C(∞): u(n)(x) exists and is continuous for all x in I regardless of n.

A:u(x) is representable by a convergent Taylor series about each x in I.

It is clear, a priori, that each restriction implies the preceding ones, and Exercise 1 will indicate that each restriction is satisfied by fewer functions than the preceding one.

What happens when we introduce [square root of -1] (or i)? We write the independent variable z in terms of two independent real variables x and y, as

(1-1) z = x + iy

and we write the dependent variable symbolically as w(z); and in terms of two real functions u(x,y) and v(x,y) we write w(z) as

(1-2) w(z) = u(x,y) + w(x,y)

Thus in real terms a complex function is a two-dimensional "surface" [x,y,u(x,y),v(x,y)] in a real four-dimensional (x,y,u,v) space. We could speak of smoothness of u(x,y) and v(x,y) in terms of tangents to this four-dimensional surface, but this type of smoothness has no interest for us. We are considering the stronger condition of differentiability of w in terms of z.

We call a function w(z) analytic in a region R if the derivative w'(z) exists for each point of the region. The function is then called analytic at each point of the region. We do not define analyticity at a point except by defining it in a surrounding region. (Synonyms for "analytic" are "regular" and "holomorphic.")

Theorem 1-1 An analytic function in R has derivatives of all orders in R and is representable as well by a convergent power series expansion at each point of the region within any circular disk centered at that point and lying in the region.

This theorem is practically the contents of an entire course in complex variables! Its importance is evident since it essentially vitiates all regularity distinctions once the derivative is presupposed. The proof of this theorem is effected by the rather ingenious well-known integration procedure which leads to the existence of power series.

Strangely enough, despite the fact that Theorem 1-1 does not mention integration, it has been proved classically only by integration. Perhaps then, rather than become overly impressed by this matter of semantics, we should do better to conclude that, from a more mature point of view, differentiation and integration are not so very different but are part of, say, some generalized study of linear operators.

EXERCISES

1 Show how to form real functions u(x) with all degrees of regularity desired above. {Hint: Show Ut(x) = [exp (– 1/x2)]/xt vanishes as the real variable x -> 0 for any positive integer t; show U0(x) [with U0 (0) = 0] serves as a function with all orders of derivative but no power series representation at the origin.}

2 Find a function w(z) = u(x,y) + iv(x,y) with a complex derivative at every point on the real axis but which is not analytic in any neighborhood of a real point. [Hint: Use U0(y).]

3 Show that [bar.f([??])] = g(z) is analytic at [??]0 whenever f(z) is analytic at z0.

1-2 Integration on Curves and Chains

We introduce integration by defining a smooth curve segment K by differentiate functions parametrized by arc length s

(1-3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here dx2 + dy2 = ds2 and l is the length in usual elementary fashion. A piecewise smooth curve (or more simply a curve) is the union of any finite set of curve segments meeting end to end, i.e., in the manner described formally as follows: If K1 and K2 are two curve segments with

(1-4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then if x1(l1) = x2(0) and y1(l1) = y2(0), we join K1 to K2, writing the sum (purely formally) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is clear how any number of segments K1 + K2 + K3 + ... + Kn can be joined to form a curve K. We shall use — K to denote the curve K parametrized in reverse fashion (with l — s replacing s if l is the length of K). We shall say qualitatively the distinction between K and — K is its orientation. Thus L = K — K would denote the tracing of K followed by the retracing of K with opposite orientation. Here K is not "zero"; it is a curve twice as long as K but one which returns to its origin.

The definition of the integral can now be effected in terms of a complex parametric form of our curve C as x + iy = z(s). If f(z) is an analytic function in a region R and if C is in the region R, then we define the integral by the Riemann sum as follows:

(1-6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here C has length l and 0 = s0< S1< ... < sM = l is a partition of l with maximum interval (sJ — Sj–1) = δ and Zj = x(si + iy(si); and ζj is an arbitrary point on the curve segment from zj–1 to zj. The existence of the limit is shown in elementary texts.

Now when we consider the integral (1-6), we become aware of the fact that retracing the path of integration nullifies the integral; for example,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We therefore introduce a purely abstract concept to make manifest this almost trivial property.

We consider the formal additive group of finite linear combinations of curves with integral coefficients. This group contains expressions such as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where K1, K2, ..., Kr need not even have a point in common. Indeed any curve Ki can have some retraced portions, or for that matter, Ki might be of the type K-K. We say that two such formal expressions constitute the same chain if they can be identified by cancellation and regrouping in obvious fashion. Thus a 0- (null) chain is defined (and it is not identified with any particular point). We can, of course, speak of an obvious correspondence between the adjoining of curve segments in (1-5) and the addition of the corresponding chains. The correspondence is not biunique; two different curves, for example, 0 and K-K, might determine the same chain. Sometimes the chains L are called "1-chains" because of the dimensionality of curves.

The purpose of all this formalism is the following idea: We can write

(1-7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and we can think of the above expression as being bilinear (or linear separately) in the differential f(z) dz and the chain C. The essence of the bilinearity is our ability to write

(1-8a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and at the same time

(1-8b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

regardless of whether or not C1 adjoins C2.

These seemingly trivial ideas will be brought to the foreground now in preparation for Part III where they play a vital role. In the meantime, our language will be sufficiently flexible for us to regard the symbol C as either a curve or a chain as the occasion requires.

The important integral estimate acquires a special symmetry when we use the bilinear form (1-7)

(1-9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where l(C) is the length of C and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the supremum of |f(z)| for z on C. Here C designates a curve (not a chain).

EXERCISES

1 Show that two piecewise smooth curves can have an infinitude of isolated points of intersection. In fact this is true even for convex segments (segments where d2y/dx2 is positive on each curve). [Hint: For –1 = ≤ x ≤ 1 consider the curves y1(x) = 20x2 + x5 sin (1/x)(x ≠ 0), y1 (0) = 0, and y2 (x) = 20x2.] Also verify that they satisfy the smoothness condition C".

2 Prove the integral estimate (1-9) by direct appeal to the definition (1-6).

1-3 Cauchy Integral Theorem

We now think of curves in terms of the boundary of a region. We define a closed curve K as one for which x(0) = x(l), y(0) = y(l) where l is the length of K. We define a simple curve as a curve with no self-intersections except in the case of a closed curve which meets itself at its end points. Symbolically, if l is the length and if 0 ≤ s1< s2 ≤ l, then (x(s1), y(s1)) = (x(s2), y(s2)) only (possibly) when s1 = 0, s2 = l.

Let us say that a region R "is bounded by a curve K" (or –K) if the set of boundary points of consists of the point set |K|. This is a concept we shall surely have to enlarge, but for the time being, it enables us to express theorems on integration.

Theorem 1-2 (Cauchy-Goursat) Let f(z) be analytic in a region R* which contains a simple closed curve C and a subregion R for which C is the boundary; then

(1-10) ∫e f(z) dz = 0

This theorem was proved by Cauchy (1825) under a (really innocuous) variant of the concept of analyticity, requiring that f'(z) not merely exist in R but also be continuous there. Goursat (1900) proved the stronger result by a proof which also had the advantage of keeping the argument in terms of complex quantities (avoiding the explicit use of f = u + iv, dz = dx + i dy, etc.).

Although the sequence of proofs is somewhat interrupted, this is a good place to insert certain corollaries to Theorem 1-2.

Corollary 1 If z1and z2are two points interior to R, as defined in the above theorem, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be defined independently of the curve joining z1to z2as long as the curve lies in R.

Corollary 2 (Morera's theorem, 1886) If f(z) is a continuous function of z in R and if

(1-11) ∫e f(z) dz = 0

about any closed curve C in R, then f(z) represents an analytic function in R.

Let us recall the proof of this last result. On the basis of Corollary 1, we can define the unique function

(1-12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where a is an arbitrarily fixed point of R and z is a variable point of R. Then F'(z) exists [= f(z) by Exercise 2] in R. Hence F(z) is analytic, and (by Theorem 1-1) so is its derivative f(z), which completes the proof. The naïvety of this device is in the characteristic vein of classical complex variable theory.

Corollary 3 If fn(z) is a sequence of analytic functions converging uniformly to a limit function f(z) in a region R* as n [right arrow] ∞, then f(z) is analytic in R*.

PROOF In any subregion R (of R*) bounded by a simple closed curve C, we apply Morera's theorem to the following result (based on uniform convergence and Theorem 1-2):

(1-13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Of course, if f(z) is analytic merely on , a closed curve, it certainly does not follow that ∫e f(z) dz = 0. The best known counterexample, of course, is

(1-14) ∫e dz/z = 2πi

where C is a simple closed curve "surrounding the origin" in the sense that the angle subtended at the origin by C increases by 2π with the parametrization. To see (1-14) as a triviality, it suffices to look upon the integrand dz/z as d log z and write

(1-15a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

meaning that if z is parametrized as z(s) = x(s) + iy(s) by a continuous variation of s from 0 to l (the length),

(1-15b) log z |C

Here we note that d log z is always single-valued as a differential despite the fact that the indeterminacy of log z can be any number of the type 2πim for m a positive or negative integer.

Equation (1-14) is important enough to be sometimes stated as a theorem. For our purposes, it is superseded by Cauchy's residue theorem (see Sec. 1-8 below). For any curve C (simple or not) and any point a not on , we note that it enables us to define the following function:

(1-16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This is called the "winding number" of C with respect to a. All we can say for now is that when C is closed, the values of N(C; a) are integers (because the indeterminacy of the argument is in multiples of 2π).

The expression N(C; a) is simultaneously a function of C and a(although the term "functional" is traditionally more appropriate). Clearly, N depends continuously on a and C unless a reaches C. This means that if a family of curves Ct (none passing through a) were parametrized continuously by parameter t, then N(Ct; a) would vary continuously with t. Finally, when C is closed, N takes on discrete values 0, ±1, ±2,.... Thus the winding number for a closed curve C about a point a remains constant, as C and a vary continuously, as long as a is never on C.

Later on we shall see that the logarithm function essentially characterizes the geometry of the plane (rather than conversely)! Indeed, we generalize the plane to "other geometrical entities" and the logarithm function to so-called "abelian integrals," but only for the plane will we be able to speak of anything like a winding number.

(Continues...)

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Excerpted from Conformal Mapping on Riemann Surfaces by Harvey Cohn. Copyright © 1967 Harvey Cohn. Excerpted by permission of Dover Publications, Inc..
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Table of Contents

Preface
PART ONE Review of Complex Analysis
  Introductory Survey
    Chapter 1. Analytic Behavior
      Differentiation and Integration
      1-1. Analyticity
      1-2. Integration on curves and chains
      1-3. Cauchy integral theorem
      Topological Considerations
      1-4. Jordan curve theorem
      1-5. Other manifolds
      1-6. Homologous chains
    Chapter 2. Riemann Sphere
      Treatment of Infinity
      2-1. Ideal point
      2-2. Stereographic projection
      2-3. Rational functions
      2-4. Unique specification theorems
      Transformation of the Sphere
      2-5. Invariant properties
      2-6. Möbius geometry
      2-7. Fixed-point classification
    Chapter 3. Geometric Constructions
      Analytic Continuation
      3-1. Multivalued functions
      3-2. Implicit functions
      3-3. Cyclic neighborhoods
      Conformal Mapping
      3-4. Local and global results
      3-5. Special elementary mappings
PART TWO Riemann Manifolds
  Definition of Riemann Manifold through Generalization
    Chapter 4. Elliptic Functions
      Abel's Double-period Structure
      4-1. Trigonometric uniformization
      4-2. Periods of elliptic integrals
      4-3. Physical and topological models
      Weierstrass' Direct Construction
      4-4. Elliptic functions
      4-5. Weierstrass' Ã function
      4-6. The elliptic modular function
      Euler's Addition Theorem
      4-7. Evolution of addition process
      4-8. Representation theorems
    Chapter 5. Manifolds over the z Sphere
      Formal Definitions
      5-1. Neighborhood Structure
      5-2. Functions and differentials
      Triangulated Manifolds
      5-3. Triangulation structure
      5-4. Algebraic Riemann manifolds
    Chapter 6. Abstract Manifolds
      6-1. Punction field on M
      6-2. Compact manifolds are algebraic
      6-3. Modular functions
PART THREE Derivation of Existence Theorems
  Return to Real Variables
    Chapter 7. Topological Considerations
      The Two Canonical Models
      7-1. Orientability
      7-2. Canonical subdivisions
      7-3. The Euler-Poincaré theorem
      7-4. Proof of models
      Homology and Abelian Differentials
      7-5. Boundaries and cycles
      7-6. Complex existence theorem
    Chapter 8. Harmonic Differentials
      Real Differentials
      8-1. Cohomology
      8-2. Stokes' theorem
      8-3. Conjugate forms
      Dirichlet Problems
      8-4. The two existence theorems
      8-5. The two uniqueness proofs
    Chapter 9. Physical Intuition
      9-1. Electrostatics and hydrodynamics
      9-2. Special solutions
      9-3. Canonical mappings
PART FOUR Real Existence Proofs
  Evolution of Some Intuitive Theorems
    Chapter 10. Conformal Mapping
    10-1. Poisson's integral
    10-2. Riemann' s theorem for the disk
    Chapter 11. Boundary Behavior
    11-1. Continuity
    11-2. Analyticity
    11-3. Schottky double
    Chapter 12. Alternating Procedures
    12-1. Ordinary Dirichlet problem
    12-2. Nonsingular noncompact problem
    12-3. Planting of singularities
PART FIVE Algebraic Applications
  Resurgence of Finite Structures
    Chapter 13. Riemann's Existence Theorem
    13-1. Normal integrals
    13-2. Construction of the function field
    Chapter 14. Advanced Results
    14-1. Riemann-Roch theorem
    14-2. Abel's theorem
    Appendix A. Minimal Principles
    Appendix B. Infinite Manifolds
    Table 1: Summary of Existence and Uniqueness Proofs
    Bibliography and Special Source Material
    Index