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Automorphic Functions 1ST Edition
Automorphic Functions 1ST Edition
Lester R FordCategories:
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1929
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MGH
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MCGRAW HILL PUBLISHING COMPANY
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english
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345
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AUTOMORPHIC FUNCTIONS BY LESTER R. FORD, Ph.D., Assistant Professor of Mathematics in the Rice Institute This volume was written during the tenure of a National Research Fellowship First Edition' /> v У "** . ^ \ \? '"^V.l.e;  vac *' ^' McGRAWHILL BOOK COMPANY, Inc. NEW YORK: 370 SEVENTH AVENUE LONDON: 6 & 8 BOUVERIE ST., E. C. 4 1929 AUTOMORPHIC FUNCTIONS Copyright, 1929, by the McGrawHill Book Company, Inc. PRINTED IN THE UNITED STATES OF AMEBICA THE MAPLE PRESS COMPANY, YORK, PA. PREFACE In 1915, the author published in the series of Edinburgh Mathematical Tracts a brief introduction to "Automorphic Functions." This booklet has been long out of print. Except for this little volume, no book on the subject has ever appeared in English. This is regrettable, in view of the importance of the subject to those whose interests lie in the field of Functions of a Complex Variable and of its numerous contacts with other domains of mathematical thought. It has been the author's aim in the earlier chapters to lay the foundations of the theory with all possible rigor and simplicity. The introduction and use of the isometric circle (the name was suggested by Professor Whittaker) have given the theory of linear groups a simplicity it has not had hitfierto.^ The fundamental region which results was given by J. I. Hutchinson, in 1907, and independently by G. Humbert, in 1919. But the interesting properties of the circle itself and its utility in the derivation of the theory seem to have escaped the attention of workers in the field. It may be stated here, that much of the material involving the isometric circle embodies researches of the author which have not appeared elsewhere in published form. In the later chapters, the author's task has been largely a matter of selection of material and method of treatment. Here, also, the use of the isometric circle has often led to a simplification of the proofs. The material included has been, perhaps, a matter of the author's personal taste. Th; e classical elliptic modular functions deserve a place as the bestknown examples of nonelementary automorphic functions. The theory of conformal mapping has been presented at some length—from the modern functiontheory point of view—as a preparation for the theories of uniformization which follow. The chapter on Conformal Mapping can be read independently of the rest of the book. In the final chapter, the connection between automorphic functions and differential equations is brought out. ν VI PREFACE This treatment, which is necessarily brief, leads up to the triangle functions. % The connection between groups and nonEuclidean geometry has not been treated, as it now seems of less importance, in view of the way the foundations have been laid. A chapter on Abelian Integrals, treated in the light of uniformization, would have been of interest; however, the subject has not been worked out to any extent. This chapter should probably not be written until some geometer sets up a product for the prime function for the Fuchsian group of the first kind. Finally, the book might have been improved by a more extensive use of the theory of normal families of functions. It is now almost fifty years since Poincare created the general theory of automorphic functions, in a brilliant series of papers in the early volumes of Acta Maihematica. Since that time, the subject has had a steady growth. The material in the present volume will be found to spring very largely from researches of the past twenty years, either in content or in method of treatment. The theory of uniformization rests on the papers of Koebe and Poincare, published in 1907. The first rigorous prqpf of the possibility of mapping one plane simply connected region upon another was presented by Osgood, in 1901, but the treatment of this and similar problems by functiontheory methods came a dozen years later. Area theorems and other aspects of mapping are of still later date and are active subjects of investigation today. The foundations of the theory of groups, as previously stated, are based on the author's studies during the past few years. The author wishes to thank Prof. Ε. Τ. Whittaker for various kindnesses while at the Mathematical Laboratory of the University of Edinburgh. He is indebted similarly to Profs. Otto Holder and Paul Koebe, of the University of Leipzig. He received much profit from the lectures of Professor Holder on "Elliptic Modular Functions" and from those of Professor Koebe on "Uniformization." So much of the material in the latter half of the book was gleaned from the published papers of Professor Koebe that specific references have often not been given and attention is called here to that fact. The author acknowledges a debt of long standing to Prof. W. F. Osgood, whose inspiring teaching first aroused an interest in the subject dealt with in this volume and whose " Funktiorientheprie " has been a mine of usefulness. PREFACE vii The preceding gentlemen are in no wise responsible for such defects as the book possesses. Their encouragement and their suggestions have been valuable; but they are busy men, and the author has not presumed to ask them to read the manuscript. The only assistance in the task of writing the book was from Mr. Jack Kronsbein, a student of the University of Leipzig, who, as a labor of friendship, typed most of the manuscript and checked many of the formulae. The author's chief debts are to the National Research Council, whose grant of a fellowship made the writing of the book possible, and to the Rice Institute, which gave him leave of absence. L. R. F. Houston, Texas. June, 1929. CONTENTS Page Preface ν Chapter I Linear Transformations 1. The Linear Transformation 1 2. Symbolic Notation 4 3. The Fixed Points of the Transformation 6 4. The Linear Transformation*and the Circle 8 5. Inversion in a Circle 10 6. The Multiplier, К 15 7. The Hyperbolic Transformation, К = A 18 8. The Elliptic Transformation, К = eie 19 9. The Loxodromic Transformation, К = Aeie 20 10. The Parabolic Transformation 21 11. The Isometric Circle 23 12. The Unit Circle 30 Chapter II Groups of Linear Transformations 13. Definition of a Group. Examples 33 14. Properly Discontinuous Groups 35 15. Transforming a Group 36 16. The Fundamental Region 37 17. The Isometric Circles of a Group. 39 18. The Limit Points of a Group. ... 41 19. Definition of the Region R . . . . 44 20. The Regions Congruent to R 44 21. The Boundary of β 47 22. Example. A Finite Group . . 49 23. Generating Transformations 50 24. Cyclic Groups 51 25. The Formation of Groups by the Method of Combination.... 56 26. Ordinary Cycles 59 27. Parabolic Cycles 62 28. Function Groups 64 Chapter III Fuchsian Groups 29. The Transformations 67 30. The Limit Points 67 ix χ CONTENTS Page 31. The Region R and the Region RG 69 32. Generating Transformations 71 33. The Cycles . 72 34. Fuchsian Groups of the First and Second Kinds 73 35. Fixed Points at Infinity. Extension of the Method 75 36. Examples 78 37. The Modular Group 79 38. Some Subgroups of the Modular Group 81 Chapter IV Automorphic Functions 39. The Concept of the Automorphic Function 83 40. Simple Automorphic Functions 86 41. Behavior at Vertices and Parabolic Points 88 42. The Poles and Zeros 91 43. Algebraic Relations , . . . ' 94 44. Differential Equations 98 Chapter V The Poincare Theta Series 45. The Theta Series 102 46. The Convergence of the Series 104 47. The Convergence for the Fuchsian Group of the Second Kind . 106 48. Some Properties of the Theta Functions 108 49. Zeros and Poles of the Theta Functions 112 50. Series and Products Connected with the Group 115 Chapter VI The Elementary Groups I. The Finite Groups 51. Inversion in a Sphere 117 52. Stereographic Projection 119 53. Rotations of the Sphere 120 54. Groups of the Regular Solids 123 55. A Study of the Cube 124 56. The General Regular Solid 127 57. Determination of All the Finite Groups 129 58. The Extended Groups 136 II. The Groups with One Limit Point 59. The Simply and Doubly Periodic Groups 139 60. Groups Allied to the Periodic Groups 140 61. The Automorphic Functions 144 III. The Groups with Two Limit Points 62. Determination of the Groups 146 CONTENTS xi Chapter VII The Elliptic Modular Functions Page 63. Certain Results from the Theory of Elliptic Functions 148 64. Change of the Primitive Periods 150 65. The Function J(r) 151 66. Behavior of J (τ) at the Parabolic Points 153 67. Further Properties of J (τ) 155 68. The Function λ (τ) 157 69. The Relation between λ (τ) and J (τ) 159 70. Further Properties of λ (r) 160 Chapter VIII Conformal Mapping 71. Conformal Mapping ... 164 72. Schwarz's Lemma 165 73. Area Theorems 167 74. The Mapping of a Circle on a Plane Finite Region 169 75. The Deformation Theorem for the Circle 171 76. A General Deformation Theorem 175 77. An Application of Poisson's Integral 177 78. The Mapping of a Plane Simply Connected Region on a Circle. The Iterative Process 179 79. The Convergence of the Process 183 80. The Behavior of the Mapping Function on the Boundary.... 187 81. Regions Bounded by Jordan Curves 198 82. Analytic Arcs and the Continuation of the Mapping Function across the Boundary 201 S3. Circular Arc Boundaries 202 84. The Mapping of Combined Regions 203 85. The Mapping of Limit Regions 205 86. The Mapping of Simply Connected Finitesheeted Regions . . . 213 87. Conformal Mapping and Groups of Linear Transformations . . 216 Chapter IX Uniformization. Elementary and Fuchsian Functions 88. The Concept of Uniformization 220 89. The Connectivity of Regions 221 90. Algebraic Functions of Genus Zero. Uniformization by Means of Rational Functions 229 91. Algebraic Functions of Genus Greater than Zero. Uniformization by Means of Automorphic Functions 233 92. The Genus of the Fundamental Region of a Group 238 93. The Cases ρ = 1 and ρ > 1 239 94. More General Fuchsian Uniformizing Functions 241 95. The Case ρ = 0 245 Xll CONTENTS Page 96. Whittaker's Groups 247 . 97. The Transcendental Functions t 249 Chapter X Uniformization. Groups of Schottky Type 98. Regions of Planar Character 256 99. Some Accessory Functions 258 100. The Mapping of a Multiply Connected Region of Planar Character on a Slit Region 262 101. Application to the Uniformization of Algebraic Functions. . . . 266 102. A Convergence Theorem 267 103. The Sequence of Mapping Functions 271 104. The Linearity of Tn.' 273 105. An Extension .· 278 106. The Mapping of a Multiply Connected Region of Planar Character on a Region Bounded by Complete Circles 279 Chapter XI Differential Equations 107. Connection with Groups of Linear Transformations 284 108. The Inverse of the Quotient of Two Solutions 287 109. Regular Singular Points of Differential Equations 293 110. The Quotient of Two Solutions at a Regular Singular Point . . 296 111. Equations with Rational Coefficients 299 112. The Equation with Two Singular Points 303 113. The Hypergeometric Equation 303 114. The RiemannSchwarz Triangle Functions 305 115. Equations with Algebraic Coefficients 308 A Bibliography of Automorphic Functions 311 Author Index 325 Subject Index 327 AUTOMORPHIC FUNCTIONS CHAPTER I LINEAR TRANSFORMATIONS 1. The Linear Transformation.—Let ζ and z' be two complex numbers connected by some functional relation, z' = f(z). Let the values of ζ be represented in the customary manner on an Argand diagram, or гplane, and the values of z' be represented on a second Argand diagram, or г'plane. To each point ζ of the first plane for which the function is defined there correspond one or more values of z' by virtue of the functional relation. To points, curves, and areas of the 2plane there correspond, usually, points, curves, and areas in the г'plane. We shall speak of the configurations in the 2plane as being transformed by the functional relation into the corresponding configurations in the z'plane. We shall find it convenient to represent z' and ζ on the same Argand diagram, rather than on different ones. Then the functional relation transforms configurations in the 2plane into other configurations in the zplane. In what follows but one plane will be used unless the contrary is stated. The whole theory of automorphic functions depends upon a particular type of transformation, defined as follows: Definition.—The transformation , az + Ъ (лл г = —τ—7> (1) cz + a where а, Ъ, с, d are constants and ad — be 7^ 0, is called a linear transformation.1 The present chapter will be devoted to a study of this fundamental transformation. 1 This is more properly called a "linear fractional transformation"; but we shall use the briefer designation. It is also called a "nomographic transformation." If ad — be = 0, the equation reduces to z' — constant; but this case is without interest. 1 2 LINEAR TRANSFORMATIONS [Sec. 1 The quantity ad —be is called the determinant of the transformation. It will be convenient to have always ad  be = 1, (2) the determinant in the general case becoming 1 if the numerator and denominator of the fraction in the second member be divided by ±\/(ad — be). , The second member of (1/ is an analytic function of z. The linear transformation has, therefore, the property of conformal ity; that is, when a figure is transformed, angles are preserved both in magnitude and in sign. We note that for each value of z, equation (1) gives one and only one value of z'. t There is no exception to this statement if we introduce the point at infinity. Thus if с ^ 0, ζ = — d/c is transformed into z' = со, and ζ = со into ζ' = а/с; if с = 0, ζ = со is transformed into ζ' — со. Let equation (1) be solved for z: This transformation which, applied after the transformation (1) has been made, carries each configuration back into its original position is called the inverse of the transformation (1). We note that (3) is a linear transformation; hence, Theorem 1.—The inverse of a linear transformation is a linear transformation. We note that (3) is forme φ from (1) by interchanging a and d with a change of sign. When formed in this way the determinant is the same as in (1). We see from (3) that to each value of zf there corresponds one and only one value of z. We have, then, the following result: Theorem 2.—The zplane is transformed into itself in a oneto one manner by a linear transformation. Moreover, the linear transformation is the most general analytic transformation which has the property stated in Theorem 2. We shall prove first the following theorem: Theorem 3.j—If, except for a finite number of points, the plane is mapped in a onetoone and directly conformal manner upon a plane region, the mapping function is linear. Let z' = f(z) be such a mapping function; and let qu q2, . . . , <?7i(=0C)) be the excepted points. Owing to the conformality, /(z) is analytic except at the isolated points qi, . . . , qn. Now qi Sec. 1] THE LINEAR TRANSFORMATION 3 is not an essential singularity, else the function takes on certain values an infinite number of times in the neighborhood of the point, which is contrary to hypothesis. Hence, f(z) either remains finite in the neighborhood of #г·, and hence is analytic there if properly defined, or has a pole. So f(z) is a rational function of z. A rational function which is not a constant takes on every value m times, where m is the number of its poles. Since f(z) takes on no value twice, it has a single pole of the first order. If the pole is at a finite point qk we may write '  A*. + Ao = ig + ^^jg?, Al * o. (4) г — qk ζ  qk If the pole is at infinity, we have z' = A,z + Ao, Αι И 0. (4') In either case the function is linear. Corollary 1.—The most general onetoone and directly con formal (where conformality has a meaning) transformation of the plane into itself is a linear transformation. We have not defined conformality when one of the points involved is the point at infinity. Excepting the point ζ = <x> and the point of the гplane which is carried into z' = <x>; the transformation is to be conformal. Theorem 3 then applies. Corollary 2.—The most general onetoone and directly con formal transformation of the finite plane into itself is the linear transformation z' = A\Z + A0. > We shall now consider the successive performance *of linear transformations. After subjecting the гplane to the transformation (1) let a second linear transformation αζ'+β Z yz' + δ {δ) be made. Expressing г" as a function of z, we have az + Ь ζ" = a°z + d = (afl + frOg + аЪ + № (μ az + Ъ , t (ya + bc)z + yb + Bd W Making the transformation (1) and then making (5) is equivalent to making the single transformation (6). Now, (6) is a linear transformation; its determinant, in the form in which the fraction is written, is (ad — Ъс)(а8 — fiy). It is worth noting that if the 4 LINEAR TRANSFORMATIONS [Sec. 2 determinants of (1) and (5) are each unity, that of (6) is also unity without further change. If z" be subjected to a linear transformation, we conclude on combining the new transformation with (6) that the succession of three linear transformations is equivalent to a single linear transformation, and so on. We have then the following result: Theorem 4.—The successive performance of a finite number of linear transformations is equivalent to a single linear transformation. A further wellknown property of the linear transformation is expressed in the following theorem: Theorem 5.—The linear transformation leaves invariant the crossratio of four points. Let z1} z2, z3, z4 be four distinct points and let з/, z2) z3', z4' be the points into which they are transformed by (1). We shall suppose all vthe points are finite. We have 2i — Z2 whence azx + b _ az2 + b _ (ad — bc){z\ — z2) cz\ + d cz2 + d \cz\ + d)(cz2 + d) Z\ — #2 . " (csi + d)(cz2 + dY (7) (8) (ζ/  Z2')(Z3' — Zj) = (gi  Z2)(Z3  gQ. (z/  Zs)(z2  z±) (ζχ  z3)(z2  z4) If one of the points is at infinity, we make the necessary change in (8) by a limiting process. Thus, if z2 = <*> and z\ = «, (8) becomes z3r — z4' _ _z3 — z4 z2r — z/ Zi — g3 2. Symbolic Notation.—For brevity in writing and for convenience in combination, we shall represent the second member of a transformation such as (1) by the functional notation, using capital letters for the function; thus, T{z) = aZ ±1 cz + d so that (1) becomes z' = T{z). We shall speak of this as the transformation T, the argument ζ being omitted unless ambiguity migpt arise without it. If two transformations are the same, 7\(z) = T(z), this is indicated by the equation 7\ = T. Sec. 2] SYMBOLIC NOTATION 5 Let S be the transformation (5), so z" = S(z'). Then, (6) is z" = S[T(z)] = ST(z). Thus, the succession of two transformations is written as a product, the enclosing brackets of the functional notation being omitted. It should be noted that ST is the single linear transformation resulting from making first the transformation Τ and then the transformation S, the order of performance being from right to left. TS is, in general, different from ST. It is easily seen from the meaning of the symbols that the associative law of multiplication holds, U(ST) = (US)T, and there is no ambiguity in writing simply UST. In a product, any sequence of factors may be combined into a single linear transformation. The transformations equivalent to performing Τ twice, thrice, etc., are represented by T2, Тг, etc. Thus, T2(z) means T[T(z)]. The inverse of Τ is written 271; hence, from (3) The result of applying the inverse η times is represented by T~n. If we represent the identical transformation, z' = z, by 1 so that l(z) means z, we observe that positive and negative integral powers of Τ together with unity combine in accordance with the law of the addition of exponents in multiplication. The inverse of a sequence of transformations can now be written down. To find the inverse of ST we make on the plane transformed by ST the transformation $_1 followed by 771; we have (T's^iST) = t^s^t = τ*τ = i. Thus, T71^1 is the transformation which, applied after ST, carries each point back to its original position; so T~1S~1 is the inverse of ST. In a similar manner we have for any number of transformations (ST  · · UV)1 = V'lUx · · · T'SK The rule for the transposition of factors by division is easily found. Let UST = V; then, (UST)T1 = VT, or US = VT~\ and U~l(UST) = 1717, or ST = UW. 6 LINEAR TRANSFORMATIONS [Sec. 3 Then, in an equation connecting two products, the first (last) factor of one member can be transferred to the beginning (end) of the other member by changing the sign of its exponent. Thus, symbolic division is permissible, provided the proper order of performing the operations is followed. For example, the inverse W of ST · · · UV is a transformation such that WST · · · UV = 1. By repeated division on the right we get the result given above. 3. The Fixed Points of the Transformation.—The points which are unchanged by the transformation (1) are found by setting z' — ζ in (1) and solving the resulting equation. ζ = ^Ά or cz2 + (d  a)z  Ь = 0. (9) CZ ~r~ (X Suppose, first, that c^O. Then, (9) has the two roots iu ь = <^А±^ж do) where Μ = (d  a)2 + 4bc = (a + d)2  4. (11) The second expression for Μ is derived on the assumption that ad — be = 1. We see from (1) that °o is not transformed into itself, so there are at most two fixed points. If Μ = 0, that is if a + d = ±2, there is but one fixed point, « ^ «2> If с = 0, we must have a^O, d j* О since, otherwise, the determinant would be zero. We see from (1) that <x> is then a fixed point. Solving (9) we get a finite fixed point provided a j* d. The fixed points are *1 = ;гЦ:> **= °°· (13) а — а If с = 0 and a = d, (1) takes the form ζ' = ζ + V, a translation with the single fixed point, ξ = со. In the case с = 0, we see from (11) that we have, as before, two fixed points if Μ 5* 0, and one fixed point if Μ = 0. There cannot be more than two fixed pointst unless (9) is identically zero; that is, с = 0, d = a, and Ъ = 0. Equation (1) then takes the form z' = z. Hence, Sec. 3] THE FIXED POINTS OF THE TRANSFORMATION 7 Theorem 6.—The only linear transformation with more than two fixed points is the identical transformation z' — z. By means of this theorem we are able to prove the following important proposition: Theorem 7.—There is one and only one linear transformation which transforms three distinct points, Z\y z2, z3, into three distinct points, ζ/, z2, z/. We shall prove first that there is not more than one such transformation. Let Τ be one transformation carrying zh z2, Zz into ζ/, z2y Zs', and let S be any other such transformation. Consider the transformation T^S. We have S(zi) = z/ and T\zi!) = Zl; so TiSfa) = T\zx') = z^ Hence, Zi, and similarly z2 and z3, are fixed points of the transformation Γ1£. It follows from Theorem 6 that T^S = 1; whence applying the transformation Τ to both members, S = Γ. There is, thus, not more than one transformation of the kind required. We shall prove that there is always one such transformation by actually setting it up. If none of the six values is infinite, consider the transformation defined by (г'  zi')(z2'  zz') (g  3i)(z2  z3) П4Л iz'  *,')(*!' " **) (z " *)(*i " **)' ^ an equation which expresses the equality of the crossratios (ζ'ζχ', Z2fZs) and (ζζι, ζ2ζ3). This is of the form (1) when solved for z' in terms of г. It obviously transforms zh z2, zs into z/, z2, Zz) for both members of (14) are equal to zero when ζ = Zi, z' — z<!\ they are both infinite when ζ = z2, zr = z^", and they are both 1 when ζ = z3, z' = z3'. If one of the given points is at infinity, we have but to replace the member of (14) in which that point occurs by its limiting value when the required variable becomes infinite. If Zi = oo, z2 = oo, or z3 = oo, we replace the second member of (14) by Z2 — Z3 Z — Zi Z — Z\ , , or , Z — Z2 Z\ — Z3 Ζ — Z2 respectively; and a similar change is necessary in the first member for an infinite value of ζ/, ζ2) or z3'. In any case, there is one 8 LINEAR TRANSFORMATIONS [Sec. 4 transformation with the desired property, and the theorem is established. Equation (14) is a convenient form for use in actually setting .up the transformation carrying three given points into three given points. Theorem 7 will be of great utility in our subsequent work ; to prove that two transformations are identical, we shall have merely to show that they transform three points in the same way. 4. The Linear Transformation and the Circle.—Since we are operating on the complex number ζ it will be convenient to have the equations of curves expressed directly in terms of z. If χ is the real part of ζ and iy is its imaginary part, and if we represent by ζ the conjugate imaginary of z, we have ζ — χ + iy, ζ — χ — iy. (15) From these we have x = H(* +2), У = \{г  ζ), ζζ = χ* + у\ (16) From the first two equations of (16) we can readily express the equation of any curve in terms of ζ and z. We shall now get the general equation of the circle and of the straight line. The equation A(x2 + y2) + Ъ1Х + Ъ2у + С = 0, where the constants are real, is the general equation of the circle (possibly of imaginary or zero radius) if A ^ 0, and is the general equation of the straight line if A = 0 and Ъг and b2 are not both zero. Substituting from (16) we have Azz + У2(ЪХ  ib2)z + K(bi + ib2)z + (7 = 0. Putting В = ХА(Ъ\ — ib2)j this takes the form Azz + Bz + Bz + С = 0 (17) where A and С are real. Equation (17) is the general equation of the circle if A ^ 0 and of the straight line if A = 0, В ^ 0. The center and radius of the circle are easily found. Writing (17) in the form \z + a)\z + a)= a* ' we see that the first member is the square of the distance of ζ from —B/A. Hence, (17) is a circle with center — B/A and ipp /Try radius л/ ^ Since we shall be interested only in real circles, we shall require that BB > AC. Sec. 4] THE LINEAR TRANSFORMATION AND THE CIRCLE 9 Now let us see what the circle or straight line (17) becomes when the linear transformation (1) is applied. Substituting from (3) dz' + Ъ _ dz' + Ъ ζ = cz — a cz — a we have Λdz' + b)(dz' +5) „dz' + Ъ , dg' + g , n A —τ—— \γ—ί——гч г о , h L>—^r, ζ h С = О. (18) Clearing of fractions and collecting terms, we have Add  Bed Hcd + Ccc]z'z' + [Abd + Bad + 5бс  Сафг + [АЪЯ + Bbc + Bad  Cac]z' + Abb  Bab  Bab + Cad = 0. (19) In this equation the coefficient of z'z' is real, for dd and cc are real, being each the product of a number by its conjugate; 'and Bed + Bed is real, being the sum of a number and its conjugate. Similarly the constant term is real. Also the coefficient of z' is the conjugate of the coefficient of z'. Therefore (19) is of the form (17). We have then the following result: Theorem 8.—The linear transformation carries a circle or straight line into a circle or straight line. It will often be convenient to consider the straight line as a circle of infinite radius, in which case we say briefly that a circle is carried into a circle. It is easy to see when the transform will be a straight line. The straight line is characterized by the fact that it passes through the point <*>. Hence, if the point which is carried to <*> lies on the original circle or straight line, the transform will be a straight line; otherwise the transform will be a circle. This is easily shown analytically. For the coefficient of z'z' will vanish if (dividing by cc) Чс)(1)+Чс)+Ч?)+с=* that is, if —d/c lies on the original circle or straight line. Since three points determine a circle, we can set up a transformation which carries three distinct points of the first circle into three distinct points of the second circle. Having chosen 10 LINEAR TRANSFORMATIONS [Sec. 5 the first three points, the transformed points can be selected in an infinite variety of ways (<*>3 ways, in fact); and each different selection gives a different transformation. Hence, Theorem 9.—There exist infinitely many linear transformations which transform a given circle into a second given circle. In particular, we may choose the second circle to be * the same as the first. Hence, there are <x>3 linear transformations which transform a given circle into itself. 5. Inversion in a Circle.—There is an intimate relation, as we shall now show, between the linear transformation of the complex variable and the geometrical transformation known as "inversion in a circle/' Consider a circle Q with center at К and radius r. Let Ρ be any point of the plane and construct the half line KP, beginning at К and passing through P. Let Pi be a point on the half line KP such that ΚΡλ · KP = r2; then Pi is called the " inverse of Ρ with respect to the circle Q." The relation is a reciprocal one; Ρ is the inverse of Pi. We speak of Ρ and Pi Γιο. ι. as points inverse with respect toQ. Inverse points have the property that any circle passing through Ρ and Ph the inverse of Ρ with respect to Q, is orthogonal to Q. For, let Q' be any circle through Ρ and Pi and draw KT tangent to Q', Τ being the point of tangency. We have KT* = KPX · KP = r2, whence, Τ lies on Q. The radii to Τ are perpendicular and the circles are orthogonal. We shall now get an analytic expression for the transformation. Let P, Pi, and К be the points z, zh and к in the Argand diagram. The equations of the transformations are (zi — k)(z — k)\ = r2, arg(zi — k) = arg(z — к). The first equation expresses the condition KPi · KP = r2; the second expresses the collinearity of К, Р, and Pb Since arg(z — k) = — arg(l — k), the two equations are satisfied if, and only if, (zi k){zk) = r2. (20) This is the equation of inversion ш terms of complex variables. Sec. 5] INVERSION IN A CIRCLE 11 If Q is the circle, Azz + Bz + ~Bz + С = О, (17) equation (20) becomes, on substituting the center and radius previously found, BB  AC (•■+IX,+S) which on simplification is Az& + Bz1 + Bz + С = 0, (21) We thus get the relation between ζ and its inverse Z\ from the equation of Q by substituting Z\ for ζ and leaving ζ unchanged. Solving (21) we have the explicit form of the transformation, *  ТГПГ <22> When A = 0 so that (17) is a straight line, we shall still use formulae (21) and (22) for the inversion. It is not difficult to show geometrically that when A approaches zero, Ρ and Pi attain positions such that Q is the perpendicular bisector of the segment Ρ Рь Inversion then becomes a reflection in the line Q. To show this analytically, let z2 be a point on Q. Then, z2 — z\ is the distance from z2 to z. The distance from z2 to zu using the equation of the transformation and the equation of Q (with A = 0) which z2 satisfies, is \Z2 ~ Ζχ\ ■Bz2  С Вг + С\ В ' В β (β  г2) = 22  3 . Thus all points of the line Q are equidistant from Ρ and Pi. We shall now prove the following proposition: Theorem 10.—The linear transformation carries two points which are inverse with respect to a circle into two points which are inverse with respect to the transformed circle. Let ζ and Z\ be inverse with respect to the circle (17); then (21) is satisfied. Make the transformation (1) and let z', z/, be the transformed points. We'have dz^ + b . dz' + b zi = }——, ζ = =,——; czi —a cz — a whence, substituting in (21), ЦФ > + b>(l5 + g> + B~dz/ + Ъ + ДУ + l + С  0. (czi — α)(^' — α) czi —a cz' — a 12 LINEAR TRANSFORMATIONS [Sec. 5 This equation is the same as (l8) except that z' is replaced by Z\\ hence, on simplifying we shall get (19) with z' unchanged and z' replaced by z/. But this is the condition that z' and ζ/ be inverse points with respect to the transformed circle (19).· Let us return to our study of the inversion. We see from (22) that to each ζ there corresponds one and only one Z\. Likewise to each zx corresponds one and only one z; hence, the transformation is onetoone. The fixed points of the inversion are seen, from the geometrical construction, to be the points of Q itself. This appears analytically if we set Z\ = ζ in (21). The inversion (22) can be written as the succession of the two transformations Bz2  С Z2==Z> Zl = Az7+~B' The first of these is a reflection in the real axis; the second is a linear transformation. The first preserves the magnitudes of angles but reverses their signs; the second makes no alteration. Hence, the inversion is inversely conformal. This follows also from the fact that the second member of (22) is an analytic function of z. The reflection in the real axis obviously carries circles into circles, and the succeeding linear transformation does likewise. Again the reflection carries a circle and two points which are inverse with respect to it into a circle and two inverse points; and this relation, by Theorem 10, is preserved by the linear transformation. We summarize the results in the following theorem: Theorem 11.—Inversion in a circle is a onetoone inversely conformal transformation which carries circles into circles, and carries two points inverse with respect to a circle into two points inverse with respect to the transformed circle. Since the inversion is a onetoone transformation of the plane which preserves the magnitude of the angle but changes its sign, the result of performing two inversions, or any even number of inversions, is a onetoone transformation which preserves both the magnitude and the sign of the angle. According to Theorem 3, Corollary 1, such a transformation is a linear transformation. Hence, we have the theorem: Theorem 12.—The successive performance of an even number of inversions is equivalent to a linear transformation. Sec. 5] INVERSION IN A CIRCLE 13 We shall prove that, conversely, any linear transformation is so constituted. First, let us examine some of the simpler linear transformations and find equivalent pairs of inversions. (a) The translation, z' — ζ + b. By this transformation each point of the plane is translated parallel to the line Ob (Fig. 2) a distance equal to the length of Ob. Let Li and L2 be two lines perpendicular to the line Ob and at a distance apart equal to half the length of Ob. A reflection in Li followed by a reflection in L2, the lines being designated as in the figure, is equivalent to the given translation. It is sufficient to note that three points are transformed in the proper manner (Theorem 7). We observe at once that Fig. 2. Fig. 3. the points of Lh which are unchanged by the first reflection, are translated in the desired manner by the second reflection. (b) The rotation, z' = eiez. Each point is rotated about the origin through an angle Θ. A reflection in Lx, followed by a reflection in L2, arranged as in Fig. 3, clearly rotates the points of Lx as required; hence, the two reflections are equivalent to the desired rotation. (c) The stretching from the origin, z' = Az, A > 0. Each point is transformed into a point with the same argument, but with the modulus multiplied by A. There is a stretching from the origin, or, if A < 1, a contraction toward the origin. This is equivalent to an inversion in a circle Qi with center at the origin and radius ri (Fig. 4) followed by an inversion in a circle Q2 with center at the origin and radius r2 = Гхл/А. For, if Zi, z' are the successive transforms of z, we have Ζγζ — /Ι2, ζ'ζγ — ri2A; whence, . Г\ Л. су λ Ζ . ζ' = г— = r^A · —0 = Az. Ζχ Г!2 (d) The transformation z' = — 1/z. 14 LINEAR TRANSFORMATIONS This can be written [Sec. 5 Z\ 1_ Z\ It is, thus, a reflection in the imaginary axis, ζ + ζ = 0, followed by an inversion in the unit circle, zz = 1 (Fig. 5). Consider now the general transformation (1). If с ^ 0, we can write this , _ a _ be — ad _ 1 с c{cz + d) (22') Fig. 4. Fig. 5 supposing ad — be = 1. This can be written as the following sequence of transformations: Zi z + d Zz c'zh zA =  1 z4 + С ' Z3 С The first, third, and fourth of these transformations have each been shown to be equivalent to a pair of inversions. The second can be broken into two, putting ς2 = Aeie, A > 0, z2 = eiezh Zz = Az2, each of which is equivalent to a pair of inversions. If с = 0, the transformation has the form zf = az + ч£. Putting a = Аег0, this is equivalent to the following sequence: Z\ = eiez, z2 = Azi, zf = z2 + β. We have proved the converse of Theorem 12. Theorem 13.—Any linear transformation is equivalent to the successive performance of an even number of inversions in circles. A linear transformation can be expressed as the sequence of inversions in an infinite number of ways. Later, we shall show Sec. 6] THE MULTIPLIER, К 15 that any linear transformation is equivalent to four suitably chosen inversions, and that a transformation of the class subsequently called " nonloxodromic " can be expressed as a succession of two inversions. Let us consider now the most general onetoone inversely conformal transformation of the plane, z' = V(z), where, of course, V is not an analytic function of z. If we first make a reflection in the real axis, zx = z, and then apply the preceding, we have a onetoone directly conformal transformation, and hence, by Theorem 3, Corollary 1, a linear transformation of z. That is, VM  Г»  S# and ,  γ® = ±. (23) Inversion is seen to be a special case of this general transformation. Equation (23) is a reflection followed by a linear transformation. So we can state the following general result: Theorem 14.—The most general onetoone conformal transformation of the plane into itself is equivalent to a succession of inversions in circles. The transformation is directly or inversely conformal according as the number of inversions is even or odd. 6. The Multiplier, K.—We have already noted a separation of linear transformations into two classes. Putting aside the identical transformation z' — i) a transformation has either two fixed points or one fixed point. The number of fixed points and the behaviour of the transformation with reference to the fixed points furnish a useful basis of classification of linear transformations. We treat, first, the larger class with two fixed points. Suppose, first, that in the transformation (1) с ^ 0. The finite points ξ\, ξι [Equation (10)] and the point <x> are carried into ξι, ξ%, and а/с, respectively. Hence, from (14), the transformation may be written (/ω0ι.) *~h' 16 LINEAR TRANSFORMATIONS [Sec. 6 or f^Li* = g*=b (24) ζ — ξ2 ζ — ξ2 whore К = а~~^. (25) а — сЬ К is called the "multiplier" of the transformation; its value, as we shall see, determines the character of the transformation. To get an expression for К in terms of the coefficients of the transformation, we form the following symmetric function of ξι and ξ2: 7 , 1 о  c{i о  eft Α α — βξ2 « — ci Jla?  2ac(b + fa) + c2({i2 + fa2) a2 * ac({i + Ы + с2Ы2 Since fa and ξ2 are the roots of cz2 + (d  a)z 6 = 0, (9) we have ad 6 μ, Μ_(α~ ^)2 + 2Ъс ξΐ "Η ξ2 ~ ~ > ςΐζ2 — — > ξΐ Γ ?2 —  ο с с cz Making these substitutions and simplifying, we have К +  = (α + rf)2 ~ 2ad + 2bc i£ ad — be or, if ad — be — 1, Я + ^ = (« +^)2 ~ 2. (26) We observe from (26) that the value of К depends solely upon the value of a + d. If we replace К by 1/K, equation (26) is unaltered; hence, the two roots of (26) are reciprocals. The particular root to be used in (24) depends upon which fixed point is called ξι and which ξ2. Another simple equation satisfied by К is got from (26) by transposing the 2 and extracting the square root: VK+—/==a + d. (27) VK We now make the change of variables Ζ = G(z) = I ~_ , Z> = G(z') = ?^=, (28) transformations which carry ξι and ξ2 to 0 and °o, respectively. Sec. 6] THE MULTIPLIER, К 17 Then, (24) takes the form Z' = KZ. (29) Call this transformation К so that K{Z) = KZ, and we have for the original transformation z> = G~\Z') = G~^K(Z) = GiRGiz). Writing for the original transformation z' — T{z), we have Τ = G'KG, whence К = GTG~\ (30) Let F be any configuration (point, circle, region, or what not) and let F be carried into F' by K. Operating with Τ on G~X{F), we have TG~\F) = G^KGGKF) = G*K(F) = G~l{Ff)) that is, Τ carries G~l{F) into G~l{F') We shall use this fact in the following manner: We shall investigate the simple transformation (29) and, having found how configurations are transformed, we shall carry the results back to the case of ζ and z' by applying G~l. The transformation with с = 0, , az + Ъ ζ — , , ad = 1 a can be put in the form (29) and treated similarly. One fixed point is oo, the other is ξι = ^ . We find easily that г.) (3D d  ξι = К =  a K(z a d where I (32) Putting Ζ = G(z) = ζ  ξΐ9 Ζ' = G{z') =г'  ξΐ9 transformations which carry ^ and & to 0 and oo, respectively, we have (29) as before. We have also jf+l_5+^« ^i+A2 _ (« + d)'  2ad _ К a a ad ad hence, for this case also К satisfies equations (26) and (27). An advantage in writing a transformation in terms of К lies in the ease with which powers of the transformation can be written down. If the transformation (1) be repeated η times, the equivalent single transformation becomes rapidly complicated if expressed in terms of а, Ъ, с, d. But, if we use (24) or (31), we 18 LINEAR TRANSFORMATIONS [Sec. 7 have, obviously, as the result of η applications of the transformation ^r~  Я^Чг' or *'& = *"(*  ii). (33) z — ξ2 ζ — ξ2 Thus, we merely replace the multiplier К by Kn. Similarly, for the inverse we use the multiplier K~l; and for η applications of the inverse we use K~n. Writing К in terms of its modulus A (>0) and its amplitude 0, so К = Ае«, we distinguish the three classes of transformations treated in the following sections. 7. The Hyperbolic Transformation, К = A.—We assume that A 9^ 1, since, otherwise, we have the identical transformation. The transformation Z' — AZ is the stretching from the Fig. 6. origin studied in Sec. 5(c). We observe at once the following facts concerning it: (1) a straight line through the origin (that is, a circle through the fixed points 0 and oo) is transformed into itself, each half line issuing from the origin being transformed into itself; (2) the half plane on one side of a line through 0 is transformed into itself; (3) any circle with center at the origin (and hence orthogonal to the family of fixed lines) is transformed into some other circle with center at the origin; (4) the points 0 and oo are inverse points with respect to any circle with center at the origin. We now make the transformation G~x which carries 0 and oo to ξι and £2. We have, then, the following facts concerning the hyperbolic transformation: (1) any circle through the fixed Sec. 8] THE ELLIPTIC TRANSFORMATION, К = eif> 19 points is transformed into itself, each of the two arcs into which the circle is separated by the fixed points being transformed into itself; (2) the interior of a circle through the fixed points is transformed into itself; (3) any circle orthogonal to the circles through the fixed points is carried into some other such circle; (4) the fixed points are inverse with respect to each circle of (3). Figure 6 shows the two families of circles just mentioned. The way in which regions are transformed is indicated in the figure, each shaded region being transformed into the next in the direction of the arrow. From (26), we get the condition that the transformation be hyperbolic in terms of a + d. The quantity К + ^ has for real positive values of К the minimum value 2 when К = 1. Since Κ τ* 1, we have К + ^ > 2; whence, from (26), (a + d)2 > 4. Hence, in order that the transformation be hyperbolic it is necessary that a + d be real and \a + d\ > 2. That this condition is sufficient will appear presently. 8. The Elliptic Transformation, К = eie — Here, θ ^ 2ηπ. The transformation Z' = ei6Z is the rotation about the origin Fig. 7. discussed in Sec. 5(b). The straight lines and circles of the preceding section have their roles interchanged. The circle with center at the origin is transformed into itself, the interior of the circle being transformed into itself. The points 0 and oo are inverse with respect to each fixed circle. A line drawn through the origin is transformed into a line through the origin which makes an angle θ with the first line. 20 LINEAR TRANSFORMATIONS [Sec. 9 Applying the transformation G"1, which carries 0 and oo to £i and £2, respectively, we have the following facts: (1) an arc of a circle joining the fixed points is transformed into an arc of a circle joining the fixed points and making an angle θ with the first arc; (2) each circle orthogonal to the circles through the fixed points is transformed into itself; (3) the interior of each circle of (2) is transformed into itself; (4) the fixed points are inverse points with respect to each circle of (2). The character of the transformation, with θ = }/%ж is shown in Fig. 7. The shaded regions are transformed into shaded regions as indicated by the arrows. For the elliptic transformation, (26) gives (a + d)2 = 2 + e* + e~i0 = 2 + 2 cos Θ. (34) The second member is positive or zero and less than 4. Hence, a + d is real and \a + d\ < 2. From (27) we have a + d = ±Ul + e~4\ = ±2 cos . (35) If θ is commensurable with π, there will exist an integer η such that ηθ = 2т7г; and Kn = e2m7ri = 1. The result of applying the transformation η times is that each point is returned to its original position. The transformation is then said to be of period n. We shall find that the only transformations possessing this periodic property are certain of the elliptic4 transformations. We illustrate with two or three useful cases. If θ = ж the transformation is of period 2. Then Κ = βπί = —1, and a + d = +2 cos = 0. If θ = 2тг/3, the period is 3. Then К = e2™/3 = Щ1 + г'л/3), and a + d = ±2 cos = ± 1. If θ = τ/2, the period is 4. Then К = ег*'2 = i, and a + d = ± 2 cos ^ = ± γ/2. 9. The Loxodromic Transformation, ЯГ = Ae**.—Here A is positive and unequal to 1 and θ ^ 2пж. The transformation Z' = Ael'*Z can be written as the succession of the transformations Z' = eieZh Ζλ = AZ, of which one is hyperbolic, the other elliptic. There is a stretching from the origin followed by a rotation about the origin. Each Sec. 10] THE PARABOLIC TRANSFORMATION 21 circle with center at the origin is carried into another circle with center at the origin; and each half line through the origin is carried into a half line making an angle θ with the first. For the original transformation, there is a combination of the motions shown in Figs. 6 and 7. Each circular arc joining the fixed points is carried into another such arc making an angle Θ with the first. Each circle orthogonal to the circles through the fixed points is carried into another such orthogonal circle. The loxodromic transformation has, in general, no fixed circles. There is an exception when θ = ж. Then, any circular arc joining the fixed points is carried into another arc joining the fixed points and making an angle ж with the first, the two arcs thus forming a circle. Any circle through the fixed points is then carried into itself. There is, however, this difference from the preceding cases: the interior of a fixed circle is transformed into its exterior. For the loxodromic transformation, (26) gives (a + d)2 = 2 + Αβίθ + le~ie = 2 +(a + j\ cos θ + if A  j \ sin Θ. (36) In general, the second member is not real. If, however, θ = ж the second member becomes 2 — ( A + r \ which is real. But we found that A + j > 2; so in this case the second member is negative. Without exception in the loxodromic transformation a + d is a complex {nonreal) number. In the loxodromic transformation with fixed circles a + d is a pure imaginary. 10. The Parabolic Transformation.—There remains the transformation with one fixed point, which is called a " parabolic transformation." The condition that there is a single fixed point (Sec. 3) is that a + d = ±2. Then the multiplier, if defined by (26), has the value К = 1. If с τ* 0, the fixed point is ξ = (a — d)/2c. The transformation carries <*>, ξ, —d/c into a/c, ξ, οο? respectively; hence, by the use of the formula (14), it can be written , a d z „ f + Ζ с _ с 22 LINEAR TRANSFORMATIONS [Sec. 10 Subtracting 1 from each member, +* с с Now, zf  ξ ζ  ξ α _ α — d α __ α + d _ __ 1 * " с ~ ~2с с " ~ 2с ~ + с' t ι d _ α — d d __ a + d _ 1 ξ + с " ~2c~ + с ~ ~2c  ~c Hence, the transformation can be written in the form Л. = τ1, + с (37) г — ξ ζ — ξ In (37), we have +c if a + d = 2, and — с if α + d = —2 Making the change of variable, Ζ = G(z) = ^ V = (?(*') = Д, (38) ζ — ξ , ζ — ξ a transformation which carries £ to oo, we have Z' = Ζ ± с · (39) If с = 0, so that oo is the single fixed point, we have already found, in Sec. 3, that the transformation is of the form (39) without further change. We have, in fact, a = d = ±1, and ζ' = ζ ± b. (390 The transformation (39) is the translation discussed in Sec. 5(a). The plane is translated parallel to the line joining the origin to the point ± с Any line parallel to this line is (transformed into itself. The half plane on one side of a fixejd line is carried into itself. Any other straight line is carried into a parallel line. On applying G1, °o is carried to £. Parallel straight lines, intersecting at <*> only, are carried into circles intersecting only at £, and, hence, tangent at £. Hence, in the parabolic transformation: (1) any circle through the fixed point is transformed into a tangent circle through the fixed point; (2) there is a one parameter family of tangent circles each of which is transformed into itself; (3) the interior of each fixed circle is transformed into itself. The manner in which the plane is transformed is shown in Fig. 8. Each shaded region is carried in the direction of the arrow. Sec. 11] THE ISOMETRIC CIRCLE 23 It is clear from the Figs. 6 to 8 and from the reasoning on which they are based that if a linear transformation is hyperbolic, elliptic, or parabolic there passes through each point of the plane, other than a fixed point, a unique fixed circle. In particular, there is in each case a single fixed circle through oo; that is, there is one fixed straight line. This line is easily constructed; for it passes through the point — d/c which is carried to oo and the point а/с into which oo is carried. At this point we shall combine certain of the results of the latter sections into a theorem for reference. We exclude the identical transformation zf = z. Fig. 8. Theorem 15.—The transformation zf = (az + b)/(cz + d)> where ad — be = 1, is of the type stated if, and only if, the following conditions on a + d hold: Hyperbolic, if a + d is real and \a + d\ > 2. Elliptic, if a + d is real and \a + d\ < 2. Parabolic, if a + d = ±2. Loxodromic, if a + d is complex. We have proved that these conditions are necessary. That they are sufficient follows, by elementary reasoning, from the fact that they are mutually exclusive. Thus, if a + d is real and \a + d\ > 2, the transformation can be neither elliptic, parabolic, nor loxodromic, so it must be hyperbolic; and so on. 11. The Isometric Circle.—In an analytic transformation z' = f{z), a lineal element dz = z2 — Zi connecting two points in the infinitesimal neighborhood of a point ζ is transformed 24 LINEAR TRANSFORMATIONS [Sec. 11 into the lineal element dz' in the neighborhood of z'. We have dz'=f'(z)dz\ hence, the length of the element is multiplied by \f'(z)\, and the element is rotated through an angle arg f'(z). For the linear transformation z> = T{z) = ^4^ adbe = 1, (1) cz + d we have the following theorem: Theorem 16.—When the ^transformation (1) is applied, .infinitesimal lengths in the neighborhood of a point ζ are multiplied by \cz + d\~2; infinitesimal areas in the neighborhood of ζ are multiplied by \cz + d\~4. For we have df = T'(z) = ( \_ ,.y (40) dz (cz + d)2 ч whence, lengths are multiplied by IT'O21)!» or \cz + d\~2. An infinitesimal region is carried into a similar region with corresponding lengths multiplied by l^'C2)!; hence, the area is multiplied by T'(z)j2, or \cz + d\~\ We get alternative forms for T'(z) from (24) and (37). Differentiating (24) and simplifying, we obtain the first of the following results: inverting and differentiating, we find the second. For the transformation with two fixed points, For the parabolic transformation (37), we have T'W = (ί ~ξ)2' (42) the same as (41) with the value К = 1. From (41 \ we have Г'(Ь) = K: T'(b) = ~ ' (43) At ξι which is fixed, dz' = Kdz. For the hyperbolic transformation, К = A, the infinitesimal neighborhood of ξι undergoes a stretching from b; for the elliptic transformation, К = eie, there is a rotation about £x through the angle Θ; for the loxodromic transformation, К = Aeie, there is a combination of stretching and rotation. Analogous remarks apply to the neighborhood of Sec. 11] THE ISOMETRIC CIRCLE 25 £2. Since =z = je ιθ, the stretching and rotation are in the opposite sense. In the parabolic transformation, we find, on substituting ξ = (a  d}/2c into (40), that Τ (ξ) = 1. The infinitesimal neighborhood of ξ is unaltered. Lengths and areas are unaltered in magnitude if, and only if, cz + d\ = 1. If с 9^ 0, the locus of г is a circle. Writing , d\ ζ + ,,y we see that the center is j the radius is γ.· \c\ с с Definition.—The circle I, I: \cz + d\ = 1, с ^ 0, (44) which is the complete locus of points in the neighborhood of which lengths and areas are unaltered in magnitude by the transformation (1), is called the isometric circle of the transformation. The isometric circle will play a fundamental part in many of our later developments. In this section we shall investigate some of its properties. We note, first, that if с = 0, so that <*> is a fixed point, there is no unique circle with the property of the isometric circle. The derivative T'(z) is constant and equal to К (Equations (31) and (39r)). Either \K\ ^ 1, and all lengths are altered in magnitude; or \K\ = 1, and all lengths are unaltered. The latter case comprises the rigid motions—the rotations (31) and translations (39'). Theorem 17.—Lengths and areas within the isometric circle are increased in magnitude, and lengths and areas without the isometric circle are decreased in magnitude, by the transformation. d For, if ζ is within /, z + с < .,) or \cz + d\ < 1, and с \T'(z)\ > 1. A length or area within I is thus magnified in all its parts. Similarly, if г is without 7, I"(z) < 1; and a length or an area without I is diminished in all its parts. Theorem 18.—A transformation carries its isometric circle into the isometric circle of the inverse transformation. The inverse transformation, zf = ( — dz + b)/{cz — a), has the isometric circle Г: \cz  a\ = 1. (45) Its center is a/c, its radius l/c. Now Τ carries I into a circle /o without alteration of lengths in the neighborhood of any point, 26 LINEAR TRANSFORMATIONS [Sec. 11 hence T~l carries J0 back to I without alteration. But V is the complete locus of points in the neighborhood of which Τ effects no change of length; hence, J0 coincides with V. (a) Geometric Interpretation of the Transformation.—The trans * formation Τ carries / into V (Fig. 9) without alteration of any arc. Let a point Ρ on / be carried into P'. Then, if / be set down upon V so that Ρ coincides with P'', with proper orientation, corresponding points will coincide. Any sequence of an even number of inversions which will effect the proper transformation on J will be equivalent to Τ (Theorem 7). Fig. 9. As a point moves from Ρ counterclockwise around /, suppose that the corresponding point moves from Pf counterclockwise around V. Then, / can be carried into V by a rigid motion so that corresponding points coincide. But °o is fixed for a rigid motion, so с = 0; hence, this case is impossible. Consequently, as a point moves counterclockwise around / the corresponding point moves clockwise around Г. The circle / must be turned over before being applied to V. An inversion in /, leaving the points of / invariant, followed by a reflection in L, the perpendicular bisector of the line segment joining the centers, carries / into V with the desired change of order. Ρ is carried into a point Pi. A rotation with а/с fixed will carry Pi into P'. The two inversions together with the rotation are equivalent to T. Since a rotation is equivalent to two reflections (Sec. 5(&)), four inversions at most are adequate for the representation of the Sec. 11] THE ISOMETRIC CIRCLE 27 transformation. If Pf coincides with Px two inversions are sufficient. Several alternative geometric transformations are possible. Thus, instead of inverting in / and then reflecting in L we may reflect in L and then invert in I\ Or we may rotate about — d/c at the start; and so on. The preceding construction fails if / and Γ coincide, for then L is not defined. In this case a — — d, or α + d = 0; and Τ is an elliptic transformation of period two (Sec. 8). P' lies on /. An inversion in / followed by a reflection in L, the line joining — d/c to the midpoint of the arc PPf is equivalent to T. (b) The Types of Transformations.—The distance between the centers of I and V is ad с с ; the sum of the radii is 2/c. The circles will intersect, touch, or be totally exterior according as  α + d\ is less than, equal to, or greater than 2. Hence, applying Theorem 15, if Τ is hyperbolic, the isometric circles of Τ and T"1 are external; if Τ is elliptic, they intersect; if !Tis parabolic, they are tangent. If Τ is loxodromic, \a + d\ may have any value other than zero, and the isometric circles may have any relation to one another other than coincidence. A distinction between the loxodromic and the three non loxodromic transformations appears when we study the geometrical transformations which are equivalent to the transformation. Let P, Pi, P' (arranged as in Fig. 9) have the coordinates z, zh г'. Since z, —die, and ale lie on a line, ζ +  and —— have the с с с same argument, the moduli being l/\c\ and \a + d/c, respectively, hence, d _ a + d с \a + d\c Similarly, a a + d Z\ с \а + d\c Making the transformation T, using (22r), , a _ 1 _ \a + d\ __ ά+$< с c2(z + d/c) (a + d)c \a + d\c From these equations we see that zf coincides with г ι if, and only if, a + d = a + d; that is, if a + d is real. The transformation is then nonloxodromic. 28 LINEAR TRANSFORMATIONS  [Sec. 11 If the transformation is loxodromic, writing a + d = re^, φ τέ ηπ, we have , a a + d/ a\ _2г>/ а\ z' = ——J zx ) = e Ί zx )· с a + d\ c/ \ c/ To carry Zi to z', there is a rotation about а/с through the angle — 2φ. We have the result: Theorem 19.—// the transformation is hyperbolic, elliptic, or parabolic, it is equivalent to an inversion in I followed by a reflection in L; if it is loxodromic there is in addition a rotation about the center of Γ through the angle — 2 arg (a + d). Consider now the fixed points. Since Γ'(£ι) = К, ГЧЬ) = 1/K (Equation (43)), we have, if \K\ ^ 1, increase of lengths at one fixed point and decrease at the other; if \K\ = 1, there is no alteration. Hence, for the hyperbolic and loxodromic transformations one fixed point is within /, the other without; for the elliptic transformation both fixed points, and for the parabolic transformation the single fixed point, are on I. Identical statements are true of /' for similar reasons. In the elliptic transformation I and /' intersect and L is the common chord. The points of intersection are fixed for both the inversion in / and the reflection in L; hence, they are the fixed points. We found that the lineal elements issuing from the fixed point are rotated through an angle 0, where К = eie. Since an arc of / issuing from the fixed point is transformed into an arc of jP issuing from the point, it follows that I and /' intersect at the angle Θ. If a + d = 0, so that / and V coincide, the line L is the line joining the fixed points, which are then at the ends of a diameter. In the parabolic transformation, L is the common tangent to I and V at their point of tangency. The point of tangency is then the fixed point. (c) The Fixed Circles.—We consider now the nonloxodromic transformations. Each such transformation has a oneparameter family of fixed circles, including, as we found in Sec. 10, the line joining the centers of I and I'. The family of fixed circles is easily constructed. It consists of the circles with centers on L orthogonal to /. For, being orthogonal to /, such a circle is transformed into itself by an inversion in /; and a reflection in L, a diameter, transforms it again into itself. Each fixed circle is also orthogonal to /' from symmetry. Sec. 11] THE ISOMETRIC CIRCLE 29 Theorem 20.—In a nonloxodromic transformation the isometric circle is orthogonal to the fixed circles. For use later we shall prove the following theorem: Theorem 21.—Let Q be a fixed circle of a nonloxodromic transformation and I its isometric circle. Let h be the distance of a point ζ from q, the center of Q, and hf be the distance of the transformed point zf from q; then hf = h, if ζ is on I or Q; hf < h, if ζ is within both I and Q, or without both; hf > h, if ζ is within either I or Q, and without the other. An inversion in / carries ζ to a point zx which is carried to z' by a reflection in L (Fig. 10). Obviously, the reflection does not alter distances from q. The proposition hinges, then, on what happens when ζ is inverted in /. The distances of a point and its inverse from the center of a circle orthogonal to the circle of inversion is clearly independent of the orientation of the circles, and their relative magnitudes are independent of the scale used; hence, it will suffice to take for J the unit circle zz = 1 and to take q on the real axis. The equation of Q is (z for orthogonality, r2 + 1 = q2; whence, zz  q(z + z) + 1 = 0. The expression in the first member of this equation is positive for points without Q and negative for points within. Now, Д2 = (з _ q)(z _ q) = zz  q(Z + χ) + 1 + r\ and, since ζγ = 1/z, <l)(z — q) = r2, where, Д'2 = Zlz,  q(zx + zi) + 1 + r2 1 — q{z + z) f zz zz + r2 whence, h2  h'2 = [zz  l][zz  q(z + z) + 1] zz The theorem follows immediately from this equation. If ζ is within both circles or without both circles, the factors in the 30 LINEAR TRANSFORMATIONS [Sec. 12 numerator of the second member are both negative or both positive, and Ы < h; if г is within one circle and without the other, the factors differ in sign, and Ы > h; if ζ is on one circle, one factor is zero, and Ы — h. The following theorem relative to the fixed straight line is easily seen to hold: Theorem 22.—In a nonloxodromic transformation let к and kr be the distances of ζ and z', respectively, from the fixed straight line M; then к' = к if ζ is on I or on M; otherwise, к' > к if ζ is within I and к' < к if ζ is without I. 12. The Unit Circle.—We shall, subsequently, have much to do with sets of linear transformations which have one fixed circle in common. It will usually be convenient to take as the common fixed circle some simple circle such as the real axis or the unit circle with center at the origin. It is this latter circle, which we shall henceforth designate by Q0, that we shall study in this section. We proceed to find the conditions on the constants in (1) in order that Q0 be a fixed circle. The equation of Qo is zz  1 = 0. (46) The transform of Q0 by (1) is, from Equation (19), (dd  cc)z'z' + (bd + ac)zf + (bd + ac)zf + bb  aa = 0. This circle is identical with Q0 if, and only if, — bd + ac = 0, — bd + ac = 0. (a) dd — cc = aa — bb ^ 0. (6) Each equation in (a) is a consequence of the other; from the second, b _ a с d say;then b = Xc, a =* \d. Substituting.in (b) dd — cc = \\(dd — cc) ^ 0, whence, λλ = 1. From ad — be = 1, we have \(dd — cc) — 1; hence, λ is real, so λ = ±1. The sign of λ depends upon the sign of dd — cc. If the interior of .Q0 is transformed into its interior, the point — d/c which is carried to °o is outside the Sec. 12] THE UNIT CIRCLE 31 circle and — d/c\ > 1; so dd — cc > 0 and λ = 1. We have, then, b = с, а = d, d = a. These values obviously satisfy the conditions (a) and (b). We have the following result: Theorem 23.—The most general linear transformation carrying Qo into itself and carrying the interior of Qo into itself is the transformation г' = ^Jl ой  cc = 1. (47) cz + a The most general linear transformation carrying Q0 into itself and carrying the interior of Q0 into its exterior is found similarly. Then, — d/c is within Q0 and dd — cc < 0; so that λ = — L The most general transformation is the resulting loxodromic transformation z' = _, cc — ad = 1. (48) cz — a The transformation (47) maps the interior of Qo in a oneto one and directly conformal manner on itself. It is a remarkable fact, which we shall now prove, that it is the most general such transformation. We first prove the following proposition: The most general transformation which maps the interior of Qoina onetoone and directly conformal manner on itself and which leaves the origin fixed is a rotation about the origin. Let zr = f(z) be such a transformation. Owing to the con formality f(z) is analytic in Qo. Further, /(z) < 1 when \z\ < 1, since an interior point is carried into an interior point. Since z' = 0 when ζ = 0, f(z) has a zero at the origin; hence f(z)/z is analytic in Qo. Consider now \f(z)/z\ in a circle Q' with center at the origin and radius r < 1. Since the absolute value of a function which is analytic in and on the boundary of a region takes on its maximum value on the boundary, we have, since \z\ = r on Q'\ № < l in Q'. r Since r may be taken as near to 1 as we like, we have № Ζ < 1 in Q0. Considering the inverse function, ζ = φ(ζ'), we have by the same reasoning \z/z'\ < 1 in QQ. Consequently, \z'/z\ = 1 in Qo, 32 LINEAR TRANSFORMATIONS [Sec. 12 whence z'/z = eia. But, if the absolute value of an analytic function is constant, so also is its argument; hence a is constant. We have, thus, zf = eiaz, a rotation about the origin. We shall now remove the restriction that the origin be fixed. Let z' = f(z) map the interior of Qo on itself in a onetoone and directly conformal manner, and let/(0) = z0. Let z' = S(z) be a linear transformation of the form (47) such that S(0) = z0. (It is easy to determine the constants of (47) so that c/a = 20 < 1.) If we make the transformation / and then make #1, the interior of Q0 is carried into itself and the origin is fixed. Hence, >S_1/ = U, a rotation; and / = SU. We thus have a linear transformation. Since it carries the interior of Qo into itself, it is of the form' (47). Theorem 24.—The most general transformation which maps the interior of Qo in a onetoone and directly conformal manner on itself is the linear transformation (47). The proof of the following more general theorem is now easily made. Theorem 25.—The most general transformation which maps the interior or exterior of one circle in a onetoone and directly conformal manner upon the interior or exterior of another circle is a linear transformation. Let z' = f{z) carry the interior or exterior of Qi into the interior or exterior of Q2 in the manner stated. Let Si and S2 be linear transformations carrying Qi and Q2, respectively, into Qo, the interior or exterior of each which is involved in the mapping being carried into the interior of QQ. Then, the sequence of transformations, Si1, followed by /, followed by >S2, carries the interior of Q0 into itself, and is equivalent to a linear transformation Τ of the form (47). StfSr1 = T, or/= S^rSi. The transformation is thus a linear transformation. CHAPTER II GROUPS OF LINEAR TRANSFORMATIONS 13. Definition of a Group. Examples.—The automorphic function depends for its definition on a set of linear transformations called a "group." In the present chapter we shall make a study of groups of linear transformations, after which we shall be in a position to pass to the definition of the automorphic function and to a study of its properties. Definition.—A set of transformations, finite or infinite in number, is said to form a group if, (a) the inverse of each transformation of the set is a transformation of the set; (b) the succession of any two transformations of the set is a transformation of the set. The definition applies to all kinds of transformations, but we shall be concerned only with sets of linear transformations. The two group properties, expressed in' symbolic notation, are: (a) if Τ is any transformation of the set so also is T~l\ (b) if S is a transformation of the set, not necessarily different from T, so also is ST. It follows by a repeated application of (6) that the transformation equivalent to performing any sequence of transformations of a group belongs to the group. In particular, all positive and negative integral powers of a transformation Τ of the group belong to the group. Also T~1T( = 1) belongs to the group; that is, every group contains the identical transformation, z' — z. Given a set of linear transformations, we may test whether or not it constitutes a group by applying (a) and (b) to the transformations of which it is composed. There are, however, certain cases in which the group properties obviously hold. For example, if the set consists of all linear transformations which leave some configuration F in the zplane invariant, then the set is a group; for, clearly, the inverse of any transformation or the successive performance of any two transformations will leave F invariant and, being themselves linear transformations, will 33 34 GROUPS OF LINEAR TRANSFORMATIONS [Sec. 13 then belong to the set. Thus, all linear transformations with a common fixed point constitute a group. All linear transformations of the form (47), Sec. 12, leaving Qo and its interior invariant, form a group. ' The set of all linear transformations which carry a given regular polygon into itself, consisting of certain rotations about its center, form a group. Similarly, the set of all linear transformations which leave invariant some function of ζ constitute a group. For example, all linear transformations z' — T(z) such that sin z! = sin ζ form a group. Such transformations as z' — ζ + 2π, ζ' = г + 4тг, ζ' = 7г — ζ, etc., belong to this group. It is by virtue of this property, as we shall see later, that sin ζ is called an "automorphic function." Given a set of linear transformations Tb T2, . . ., Tn, we may form a group containing them in the following way: Let the set contain the given transformations, their inverses, and the transformations formed by combining the given transformations and their inverses into products in all possible ways. Then it is easily seen that the inverse of any transformation or the product of any two is itself some combination of the given transformations and their inverses and, consequently, is included in the set. Hence, the whole set forms a group. The group is said to be "generated" by the transformations Th T2, . . . , Tn, and the transformations are called "generating transformations" of the group. Examples.—The following are a few examples of wellknown groups, some of which will be discussed later. 1. A Group of Rotations about the Origin.—The m transformations, z' = z, e2iri/mz, e^{tmz, . . . , e2(ml)«/m2j form a group. They are the rotations about the origin through multiples of the angle 2ir/m. The group is generated by the transformation г' = e2irl^mz. 2. The Group of Anharmonic Ratios.—The six transformations 1 1_ z_ 1 ζ Z ~ г> ζ l z>* 1  г г"' ζ  1 form a group. It can be verified by forming the inverses and by combining the transformations that both group properties are satisfied. The group is so named for the reason that if ζ is any one of the anharmonic ratios of four points on a line, the six anharmonic ratios are given by the transformations of the group. 3. The Group of the Simply Periodic Functions.—The set ζ' = ζ + mo>, where ω is a constant different from zero, and m is any positive or negative integer or zero, forms* a group. The group is generated by the transformation г' = г + ω. Sec. 14] PROPERLY DISCONTINUOUS GROUPS 35 4. The Group of the Doubly Periodic Functions.—The set zf = ζ + πιω + τη'ω', where ω and ω' are constants different from zero and the ratio ω'/ω is not real, and where m and m' are any positive or negative integers or zero, forms a group. It is generated by the transformations ζ' = ζ + ω, ζ' = ζ + ώ'. The restrictions on ω and ω' are not necessary for establishing the group properties. 5. The Modular Group.—The infinite set of transformations zf = (az + b)/(cz + d), where a, b, c, d are real integers such that ad — be = 1, constitutes a group. For, a reference to Equations (3) and (6) of Sec. 1 shows that the inverse of such a transformation and, also, the product of two such transformations are transformations with integral coefficients and of unit determinants. Since the coefficients are real, each transformation carries the real axis into itself. 6. The Group of Picard.—The set of transformations zf = (az + &)/ (cz + d)x where a, b, c, d are either real or complex integers (i.e. of the form m + m, where m and η are real integers) such that ad — be = 1, constitutes a group. The proof is as in the preceding case. 7. A Group Allied to Qq.—In a similar manner, the transformations z' = (az + c)/(cz + a), where a and с are real or complex integers such that ай — cc = 1, form a group. The transformations of this group (Theorem 23, Sec. 12) have Q0 as a fixed circle and carry the interior of Q0 into itself. 14. Properly Discontinuous Groups.—If we compare the group of the simply periodic functions, ζ' = ζ + mu, with the group of all translations, ζ' = ζ + b, where b is any constant, we observe the following difference: In the former case there is no transform of a point ζ within the distance ω of z; in the latter group we get transforms of ζ as near to ζ as we like by taking b small enough. These two groups bring out an essential distinction. Definition.—A group is called properly discontinuous in the zplane if there exists a point z0 and a region S enclosing z0 such that all transformations of the group, other than the identical transformation, carry z0 outside 8. The automorphic functions are founded on the properly discontinuous groups, and these only will appear in our subsequent study. The groups whose transformations contain continuously varying parameters, which have given rise to so many and so profound researches, play no part in the theory to which this book is devoted and will not be considered further. A group is said to contain infinitesimal transformations if there is, for some region A and any given e > 0, a transformation z' = (az + b)/(cz + d), ad — be = 1, such that for all points ζ of A we have \z' — z\ < e. It is found without difficulty that a necessary and sufficient condition for this 36 GROUPS OF LINEAR TRANSFORMATIONS [Sec. 15 is that there be transformations for which c, d — a, and b are all arbitrarily small (but not all zero, for then we have the identical transformation). Not all groups which are free of infinitesimal transformations are properly discontinuous. The group of Picard, for example, does not contain infinitesimal transformations, since c, d — a, and b are complex integers and cannot be made arbitrarily small without being all zero. It can be shown, however, that the points into which any point is carried are everywhere dense in the whole zplane. Such a group, that is, one which does not contain infinitesimal transformations and yet which is not properly discontinuous in the zplane, is called "improperly discontinuous" in the zplane. 16. Transforming a Group.—From a given group of linear transformations infinitely many other groups can be derived by applying linear transformations to the plane in which ζ and its transforms are represented. Let Τ be any transformation of the given group, and let Τ carry ζ into z'. Let a transformation G be applied, ζ and z' being transformed into zx and z/, respectively. Then Zi is carried into z/ by the transformation S where S = GTG1; (1) for GTGl{zx) = GT(z) = G{z') = z/. Let all the transformations of the original group be altered in this manner, so that to each Τ of the group there corresponds an S given by (1). We shall show that the new set of transformations forms a group. We have S'1 = {GTG'1)1 = GT~lG~\ which belongs to the set since T~l belongs to the original group. If Si = GTiG1 is a second transformation of the set, SSi = GTG^GTtf1 = GTTXG~\ and SSX belongs to the set since TTX belongs to the original group. Thus, both group properties are satisfied. Two groups whose transformations can be made to correspond in a onetoone manner, as the S and Τ transformations are paired by virtue of (1), so that the product of any number of transformations of one group corresponds to the analogous product of the corresponding transformations of the other, are said to be "isomorphic." It will often facilitate the study of a group to transform it in the manner indicated. For example, an important point can be carried to <χ>, or an important circle can be carried into the real axis or the unit circle Q0. Having found how figures are transformed by the new group, we can then carry the results back to the old by applying G~l. For if S carries a figure F into F'} Τ carries G^(F) into G~1(F/)) as we see at once from the equation Τ = G^SG. Sec. 16] THE FUNDAMENTAL REGION 37 It should be mentioned that the transformations S and Τ of (1) are of the same type, whatever G may be. Let T = az + Ъ G = az + ^ adЪс = 1, cz + d' 72 + δ' αδ — fiy = 1, and form the product in (1), using the equations (3) and (6) of Sec. 1, _ {aba + ayb  βδο + fiyd)z + αβα — а2Ъ + β20 — αβά , . ~ (γδα + 72&  52c + 75d)z + βία  ауЪ + /?5с  aid' ^ ' the determinant of S being 1. Then, ( — aba + «7 b — βδο + jifyd) + (fiya — otyb + βδβ — add) = (0 + d). It follows that i£ has the same value for $ as for Τ (Equation (26), Sec. 6). 16. The Fundamental Region.—Before proceeding to the study of the general properly discontinuous group it is desirable to introduce the important concept of the fundamental region. Definition.—Two configurations (points, curves, regions, etc.) are said to be congruent with respect to a group if there is a transformation of the group other than the identical transformation, which carries one configuration into the other. Definition.—A region, connected or notf no two of whose points are congruent with respect to a given · group, and such that the neighborhood of any point on the boundary contains points congruent to points in the given region, is called a fundamental region for the group. The accompanying figures show fundamental regions for certain simple groups. The reader is probably already familiar with some of them. For the group of rotations about a point through multiples of an angle Θ, which is a submultiple of 2r, we draw two half lines from the fixed point forming an angle Θ. The region R0 within the angle is a fundamental region. In Fig. 11, Ro is a fundamental region for the group z' = e2niri/6z. The neighborhood of any point on the boundary of Ro contains points which can he carried into the interior of Ro by a rotation through the angle ±2t/6. In Fig. 12, Ro, whose construction is evident from the figure, is a fundamental region, or period strip, for the group of the simply periodic functions, ζ' = ζ + m<*>. 38 GROUPS OF LINEAR TRANSFORMATIONS [Sec. 16 In Fig. 13, Ro is a fundamental region, or period parallelogram, for the group of the doubly periodic functions z' = ζ + πιω + m'c/. In Fig. 14, Ro, which is bounded by circles with centers at the origin and with radii 1 and A, is a fundamental region for the group of stretchings from the origin, z' = Avz. 0 Ro ω Fig. 11. Fig. 12. Attention may be called to certain properties that are common to the four fundamental regions constructed in the figures and which we shall find to be more or less generally true—to what extent will appear from later analysis—of the fundamental regions we shall use for less simple groups. We note first that the Fig. 13. Fig. 14. boundaries of Ro in each case consist of congruent curves. In Figs. 11, 12, 14, each of the two boundaries can be carried into the other by a transformation of the group. In Fig. 13, the lower boundary can be carried into the upper by the translation z' = ζ + ω', and the left into the right by ζ' = ζ + ω. Sec. 17] THE ISOMETRIC CIRCLES OF A GROUP 39 Further, the transformations connecting congruent boundaries are generating transformations of the group. The two translations just mentioned generate the group of doubly periodic functions. In Fig. 11, all transformations are formed by successive applications of the rotation z' = e27ri/6z, which carries one boundary into the other. The like fact is true of the other examples. We note that we can add to the open region R0 one, but not both, of two congruent boundaries without getting two congruent points in the region. But Ro must remain in part an open region. The region R0 and the regions congruent to it, some of which are shown in the figures, form a set of adjacent, nonoverlapping regions covering practically the whole plane. The origin in Fig. 14, however, is not in any region congruent to R0. The angle at the vertex of R0 in Fig. 11 is a submultiple of 2π\ The sum of the angles at the four congruent vertices of R0 in Fig. 13 is equal to 2π\ These facts will reappear, suitably generalized. It is clear that the fundamental region is in no wise unique. Any region congruent to Rо will serve as a region. Furthermore, we can replace any part of R0 by a congruent part and still have a fundamental region. Thus, we can subtract a part at one boundary and add a congruent part at another. In this way the character of the bounding curve can be altered freely. Theorem 1.—7/ no ^wo points of a region are congruent, the transforms of the region by two distinct transformations of the group do not overlap. Let A be a region containing no two congruent points. Suppose that two transformations of the group, S and T7, carry A into two overlapping regions. Any point z0 in the common part is the transform by S of a point zx of A and the transform by Г of a point z2 of A. If zx and z2 are different for any z0, then zx and z2 are congruent points of A, which is impossible. If 2i and z2 coincide for every z0 in the common part, S and Τ are the same transformation (Theorem 7, Sec. 3). Since a fundamental region contains no two congruent points, we can state the following useful corollary: Corollary.—The transforms of a fundamental region by two distinct transformations of the group do not overlap. 17. The Isometric Circles of a Group.—We shall now investigate the properties of the most general properly discontinuous 40 GROUPS OF LINEAR TRANSFORMATIONS [Skc. 17 group. For such a group there exists, by hypothesis, at least one point z0 such that there are no transforms of z0 in a suitably small region about z0. Let G be a transformation carrying z0 into oo ; and let the group be transformed by G as explained in Sec. 15. It is this transformed group which we shall study. There is no point congruent to <*> outside or on a circle Qp with a given center and with radius ρ suitably large. In particular, oo is not a fixed point for any transformation of the group. Hence, in any transformation Τ = (az + b)/(cz + d) we have с τ* 0, except in the case of the identical transformation. The center of an isometric circle is congruent to °o { — d/c is carried to oo by Г); hence, the centers of all isometric circles lie within Qp. (a) The Isometric Circle of the Product of Two Transformations. Certain relations between, the isometric circles of two transformations and the isometric circle of their product will be of use now and subsequently. Consider any two transformations T _ az + Ъ ~ _ az + β ad — be = 1, с ^ 0, ~~ cz + d! yz + δ' αδ — /?7 = 1, 7 ^ 0. Then, QT = (aa + №z + ctb + βά ηλ ° \ya + bc)z + yb + dd w In what follows we assume that S ^ 271 so that ST is not the identical transformation; then, the isometric circle of ST is  (7a + bc)z + yb + 8d\ = 1. (4) Represent by J«, J/, It, I/, 1st the isometric circles of S, #_1, Τ7, Τ71, S27, respectively; by tfs, g8', gt, g/, gst their respective centers; and by rs, rt, rst their radii. We have δ , a d , a (yb + δα7) * = ~' ** = V ^ = ~ 7 * = 7 *' " ~ba~^Jc)y rs = ,—,» ri = гт r** = i π~ϊ~Τ (5) ItI c Ιτα + «c From these values, we derive the following relations: 1 1 rsrt rst \ya + bc\ yc\ and whence, α δ с у W ~* 9* (6) _ уЪ + δα7 d _ у п, gst ~gt" ~ya + be + с " c(ya + 5c)' U) ι ι rs<^< Π /q\ lff  ?<l = T7 = W^\ (8) Sec. 18] THE LIMIT POINTS OF A GROUP 41 (b) The Arrangement of the Isometric Circles.—By means of the preceding equations we can derive certain simple facts concerning the isometric circles of the transformations of the group. The radii of the isometric circles are bounded. Let Τ (И1) be any transformation of the group and S (И1) a transformation of the group different from 771. Then, from (8), n2 = \gsi  gt\ · \g/  gal But each factor in the second member, being the distance between points of Qp, is less than 2p; hence, n2 < 4P2, rc < 2P. (9) The number of isometric circles with radii exceeding a given positive quantity is finite. Let /5 and J/ be any two different isometric circles with radii greater than k, a positive quantity. Then ST is not the identical transformation, and, from (6), \g/  g.\ = ψ> *■ (10) The distance between the centers of two isometric circles with radii exceeding к has thus a positive lower bound. Since the centers of all such circles lie in the circle Qp, their number must be finite. It follows from this fact that the transformations of the group are denumerable. Another consequence may be stated in the following manner: Given any infinite sequence of distinct isometric circles /i, 12, Is, . . . , of transformations of the group, the radii being rh r2, r3, . . . , then Urn rn = 0. Π = oo 18. The Limit Points of a Group.—In this section and the remaining sections of the present chapter we suppose that no transformation of the group has a fixed point at infinity, so that the isometric circles exist for all transformations except the identical transformation; and that there are no points congruent to infinity in the neighborhood of infinity. This assumption involves no essential restriction since, as we have already noted, any properly discontinuous group can be transformed into one with the properties mentioned. Consider the centers of the isometric circles. If the group contains an infinite number of transformations, the centers are 42 GROUPS OF LINEAR TRANSFORMATIONS [Sec. 18 infinite in number and, hence, have one or more cluster points. We lay down the following definitions: Definitions.—A cluster point of the centers of the isometric circles of the transformations of a group is called a limit point of the group. A point which is not a limit point is called an ordinary point It is clear that all limit points lie within or on the circle Qp of Sec. 17, since the centers of all isometric circles lie within that circle. If the group contains only a finite number of transformations, there are, of course, no limit points. Theorem 2.—In the neighborhood of a limit point Ρ there is an infinite number of distinct points congruent to any point of the plane, with, at most, the exception of Ρ itself and of one other point Since only a finite number of isometric circles have radii exceeding a given positive quantity, there are isometric circles of arbitrarily small radius in the neighborhood of P. Let Q be a small circle about Ρ and let J/, 1%, ... be an infinite sequence of isometric circles contained in Q, where the center gn' approaches Ρ as η becomes infinite. Let these be the transforms of the isometric circles I\, I2, . . . , and let Sn be the transformation carrying In into In'. The centers gn of In have at least one cluster point. Suppose first that there is such a point P' distinct from P. It will suffice to show that for any point Pi distinct from Ρ and P' there is a congruent point in Q distinct from P; for, by decreasing the region Q, we then have an infinite number of congruent points. Let In be near P' and of small enough radius that In encloses neither Pi nor P. Then, since Pi is outside In, Sn(Pi) is inside In and in Q. If Sn(Pi) is different from P, the proposition is established. If Sn(Pi) coincides with P, then Ρ is not a fixed point for Sn and _>Srt2(Pi), or Sn(P), is in Q and different from P. There remains the case that the only limit point of the centers gn is Ρ itself. Let A and A' be any two points distinct from one another and from P. ^Д and A' are outside an infinite number Of ^circles /n. For these circles the congruent points, An = Sn(A) .and An = Sn(A') are in In' and in Q. At least one of the points An and An' is distinct from P. It follows that at least one of the points A and A' has an infinite number of congruent points in Q which are distinct from P. Hence, there is not more than one point, distinct from P, which does not have an infinite number of Sec. 18] THE LIMIT POINTS OF A GROUP 43 congruent points distinct from Ρ and in Q. This establishes the theorem. Theorem 3.—The set of limit points is transformed into itself by any transformation of the group. The centers of the isometric circles consist of all points congruent to' infinity. The transform of the center of an isometric circle is the center of another isometric circle or is °o itself. Let Ρ be a cluster point of the centers gu g2, . . . Then a transformation S which carries Ρ into P' carries gh g2, . . . into Qi, #2', . . · with P' as cluster point. The points of the latter set, with the possible exception of one point at <*>) are centers of isometric circles. Hence, P' is a limit point. Furthermore, no point which is not a limit point is carried by S into a limit point, since otherwise S"1 would carry a limit point into a point not a limit point. Theorem 4.—// the set of limit points contains more than two points, it is a perfect set. A set is perfect, by definition, if it has the following two properties: (1) each cluster point of the set belongs to the set; that is, the set is closed; and (2) each point of the set is a cluster point of points of the set; that is, the set is dense in itself. That the set is closed follows at once. For, since each limit point contains an infinite number of centers of isometric circles in its neighborhood, a point at which limit points cluster has also an infinite number of centers of isometric circles in its neighborhood; hence, a cluster point of limit points is itself a limit point. To establish the second property we must show that any limit point Ρ has an infinite number of limit points in its vicinity. If Pi and P2 are two other limit points, at least one of them has an infinite number of transforms in the neighborhood of Ρ (Theorem 2). As these transforms are limit points, the second property of the perfect set is established. There are groups of transformations—the finite groups— with no limit points. Groups with a single limit point and groups with two limit points exist. A group other than these simple kinds has an infinite number of limit points. Furthermore, by a wellknown property of perfect sets, the limit points are non denumerable. Theorem 5.—// a closed set of points Σ, consisting of more than one point, is transformed into itself by all transformations of the group, then Σ contains all the limit points of the group. 44 GROUPS OF LINEAR TRANSFORMATIONS [Sec. 19 Suppose, on the contrary, that there is a limit point Ρ not belonging to Σ. Then, since Σ is closed, there is no point of Σ within a suitable neighborhood of P. Let Pi," P2 be two points of Σ. At least one (Theorem 2) has transforms in the neighborhood of P. This contradicts the hypothesis that Σ is transformed into itself. As an example of the use of the last theorem, suppose that all the transformations of the group are real. Then the real axis is always transformed into itself. It follows that the limit points all lie on the real axis. 19. Definition of the Region R.—Ρ will consist of all that part of the plane which is exterior to the isometric circles of all the transformations of the group. More accurately, a point г will belong to Ρ if a circle can be drawn with ζ as center which contains no point interior to an isometric circle. We thus rule out those limit points, if any, which are not themselves within or on an isometric circle but which have arcs of isometric circles in any neighborhood of them. Later, we shall adjoin to Ρ a part of its boundary, but for the present it shall consist only of interior points. It may be a connected region, or it may comprise two or more disconnected parts. We see from (9) that it contains all of the plane lying outside a circle concentric with Qp and of radius 3p. It is clear that no two points of Ρ are congruent. A transformation Τ carries all points exterior to It into the interior of //. Any transformation of the group, except the identical transformation, carries a point of Ρ into an isometric circle and, hence, outside P; so no point of Ρ is congruent to another point of P. 20. The Regions Congruent to R.—If we apply to Ρ the various transformations of the group, there results a set of congruent regions no two of which overlap (Theorem 1). Concerning the distribution of these regions, we have the following important theorem: Theorem 6.—Ρ and the regions congruent to R form a set of regions which extend into the neighborhood of every point of the plane. Suppose, on the contrary, that there is a point z0 enclosed by a circle Q with z0 as center and of radius r sufficiently small that Q contains neither points of Ρ nor points congruent to points of P. Then, all transforms of Q contain neither points of R nor points congruent to points of P. In particular, Q and Sec. 20] THE REGIONS CONGRUENT TO R 45 its transforms contain the centers of no isometric circles, since these are congruent to oo, which is a point of R. The interior points of Q and of its transforms are ordinary points. Since 20 is not a point of R, z0 is within or on the boundary of some isometric fcircle. Similarly, the center of each circle congruent to Q lies Within or on the boundary of some isometric circle. The proof consists in showing that there is a circle congruent to Q of arbitrarily large radius, which constitutes a contradiction. Let S be a transformation whose isometric circle Is has z0 for an interior or boundary point. The center of Is is exterior to Q. Consider the circle Qi into which S carries Q. S is equivalent to an inversion in /s followed by a reflection in a line and possibly a rotation. The magnitude of Qi is determined by the inversion. It is a matter of simple algebra to show that if the center of Q is on /s the radius of Qi is r where r8 is the radius of /s. If z0 lies within Is, rx exceeds this 'value. Since rs < 2p (Equation (9)), and r < r8 < 2p, we have ri > кг, к = 2 > 1. 1 ~ V If we apply to ft a transformation whose isometric circle has the center of Qi as an interior or boundary point, we get a circle Q2 of radius r2 where, since η > r, r2 > X—о > kr] > k2r. 1  ^ V Continuing in this manner, we prove the existence of a circle Qn congruent to Q and of radius exceeding knr. By taking η large enough, Qn will contain points of R exterior to the finite region in which the isometric circles lie. These points are congruent to points of Q, a contradiction which proves the theorem. We can now establish the following result: _ Theorem 7.—R constitutes a fundamental region for the group. 46 GROUPS OF LINEAR TRANSFORMATIONS [Sec. 20 We have already shown that no two points of R are congruent. We must show further that in any circle Q about a point Ρ on the boundary of R there are points congruent to points of R. Let z0 be a point of Q which lies in an isometric circle /. Then in the region common to Q and /, which contains no points of β, there are, by Theorem 6, points congruent to points of R. This establishes the theorem. Theorem8.—Any closed region not containing limit points of the group is covered by a finite number of transforms of R (including possibly R itself). These regions fit together without lacunce. Let A be a closed region; for example, a region bounded by a simple closed curve, having no limit points &f the group in its interior or on the boundary. Then there is V finite number of isometric circles containing points of A. For, if there is an infinite number, there are circles of arbitrarily small radius. Their centers then have a point of A as cluster point, contrary to hypothesis. A transformation S carries R into a region Rs lying in I/ the isometric circle of $_1. If I/ contains points of A, Rs may contain points of A; if A is exterior to J/, then R contains no points of A. tience, the number of regions congruent to R which lie wholly or in part in A is not greater than the number of isometric circles which contain points of A A This number is finite. Also, since there are points of R, or points congruent to points of R in the neighborhood of every point of A (Theorem 6), it follows that the regions fit together without lacunae. A is completely covered, except, of course, for the boundaries separating the various regions. Theorem 9.—Within any region enclosing a limit point of the group, there lie an infinite number of transforms of the entire region R. This theorem follows at once from the fact that there is an infinite number of isometric circles lying entirely within a given region enclosing a limit point. Each of these circles contains a region congruent to the entire region Й, and the various transforms are different regions. The preceding theorems furnish a picture of the transforms of R. R and the regions congruent to it fit together to fill up all that part of the plane which is composed of ordinary points. They cluster in infinite number about the limit points. Sec. 21] THE BOUNDARY OF R 47 21 The Boundary of R.—A point on the boundary of R is a point Ρ not belonging to R but such that in any circle with Ρ as center there are points of R. Ρ may be an ordinary point or a limit point. Obviously, Ρ cannot lie within an isometric circle. If Ρ is an ordinary point, it lies on one or more isometric circles. Since there is but a finite number of isometric circles whose arcs lie in the neighborhood of an ordinary point, a circle Q can be.drawn with Ρ as center such that Q is exterior to all isometric circles other than those which pass through P. In the most general case, a boundary point P,belongs to one of the following three categories: (α) Ρ is a limit point of the group; (β) Ρ is an. ordinary point and lies on a single isometric circle; (7) Ρ is an ordinary point and lies on two or more isometric circles. Ρ is then called a "vertex." It is desirable to include under (7) the following special case: If Ρ is the fixed point of an elliptic transformation of period two, so that, although Ρ lies on a single isometric circle, it separates two congruent arcs on the circle, we shall classify Ρ under (7) rather than (β). The advantages of this classification will appear subsequently. Concerning the boundary points of category (a), there is nothing to be added to the theorems on limit points already derived in Sec. 18. We shall show subsequently (Sec. 25) that groups exist for which the boundary points of R are all limit points. The groups of interest for our present theory, however possess ordinary boundary points also. (a) The Sides.—Consider a boundary point of category (β). Let Ρ lie on Ih and let P' on I/ be the point into which Τ carries P. We shall show that Pf is also a boundary point of category (β). We put aside the case P' = P, a situation which can arise only if It and It coincide and Ρ is a fixed point of the resulting elliptic transformation of period two; for this case has been included in (7). First, P' is within no isometric circle. Suppose Pr to be within Ia) then S magnifies lengths in the neighborhood of P\ But, since Τ carries Ρ into P' without alteration of lengths, ST magnifies lengths in the neighborhood of P. Then Ρ is within Ist y which is contrary to the hypothesis that Ρ is a boundary point. 48 GROUPS OF LINEAR TRANSFORMATIONS [Sec. 21 Second, P' does not lie on an isometric circle other than I/. For, if P' lies on Is, the transformation ST effects no alteration in lengths at P. Then Ρ is on Ist, which is contrary to the hypothesis that Ρ lies on a single isometric circle. It follows from these facts that P' is a boundary point of category (β). There is no isometric circle in the neighborhood of Ρ other than It. It is clear, then, that the points on It in the neighborhood of Ρ are likewise boundary points of category (β); so, consequently, are the congruent points on I/. We thus have as a part of the boundary an arc of It and the congruent arc on It. These arcs may consist of the entire circles or they may terminate in points of category (a) or (7). Since the arcs lie on isometric circles, they are of equal length. We have, then, the following theorem: Theorem 10.—The boundary points of R of category (β) form a . set of bounding circular arcs, or sides, which are congruent in pairs. Two such congruent sides are equal in length. (b) The Vertices.—There remain for consideration the boundary points of category (7). Through a point Ρ of this category there pass a finite number of isometric circles. Let Q be a circle about Ρ sufficiently sn\all that all isometric circles other than those through Ρ are without Q and such that any points of intersection of the circles through P, other than Ρ itself, lie without Q. The isometric circles through Ρ divide Q into a finite number of parts. One of these parts A, owing to the assumption that Ρ is a boundary point, belongs to P. The two arcs which bound A, on It and I8, say, are a part of the boundary of P. The points of these arcs other than Ρ belong to category (β). That is, at a vertex two sides of Ρ meet. Now make the transformation Τ, Ρ being carried into P' on It. By reasoning almost identical with that employed in the preceding case, we show that P' is a vertex. We can show, in fact, that Ρ and P' lie on t