# Introduction to the Theory of Analytic Functions

Macmillan and Company, Limited, 1898 - 336 ページ

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### 目次

 CHAPTER I 1 Fractions 2 Irrational numbers 3 The change of origin 5 The decimal system 6 The infinite decimal 7 Distance point and angle 9 CHAPTER II 11
 Cauchys theorem on the coefficients of a power series 129 Px does not vanish near xo 131 Criteria of identity of power series 132 CHAPTER XI 134 Remarks on Weierstrasss theorem 137 Applications of Weierstrasss theorem 139 Reversion of a power series 142 Taylors theorem for power series 144

 The axis of real numbers 12 1o Imaginary numbers and the axis of imaginary numbers 13 Strokes 15 Complex numbers and the points of a plane 16 Absolute value and amplitude of x +1i7 17 Addition of two complex numbers 18 Ratio and multiplication 20 The th roots of unity 22 The th power and th root of a stroke 23 To find the point which divides in a given ratio r the stroke from to it 24 The centroid of a system of points 25 Examples 26 THE BILINEAR TRANSFORMATION ART PAGE 20 The onetoone correspondence 27 Inverse points 28 The bilinear transformation converts circles into circles 30 Coaxial circles 31 Harmonic pairs of points 32 The double ratios of four points 34 Isogonality 36 Theory of absolute inversion 38 The bilinear transformation is equivalent to two inversions in space 42 Examples 44 CHAPTER IV 46 The logarithm in general 48 Mapping with the logarithm 50 The exponential 52 Mercators projection 54 CHAPTER V 57 The motion when the fixed points are distinct 58 Case of coincident fixed points 60 Substitutions of period two 61 Reduction of four points to a canonic form 64 Substitutions of period three 65 CHAPTER VI 67 Distinction between value when a and limit when a 68 Every sequence of constantly increasing real numbers admits a finite or infinite limit 69 Every sequence of real numbers has an upper and a lower limit 70 The necessary and sufficient condition that a sequence tends to a finite limit 71 Real functions of a real variable 72 Continuity of a function of a real variable 73 A continuous function of a real variable attains its upper and lower limits 74 Functions of two independent real variables 77 A continuous function ij attains its upper and lower limits 79 Uniform continuity of a function of one real variable 80 Uniform continuity of a function of two real variables 81 Uniform convergence to a limit 82 CHAPTER VII 84 Instance and definition of a limit 87 The derivate of a function 88 The fundamental theorem of algebra 90 Proof of the fundamental theorem 91 The rational algebraic function of x 93 CHAPTER VIII 96 Convergence 97 Simple tests of convergence for series whose terms are all positive 99 Association of the terms of a series 101 Absolutely convergent series 104 Conditionally convergent series 106 Conversion of a single series into a double series 107 Conversion of a double series into a single series 110 CHAPTER IX 113 71 Uniform convergence 115 Uniform convergence implies continuity 117 73 Uniform and absolute convergence 118 74 The real power series 119 CHAPTER X 123 The circle of convergence 125 Uniform convergence of complex series 128
 The derivates of a power series 146 Differentiation of a series of power series term by term 147 CHAPTER XII 149 Continuation of a function defined by a power series 151 The analytic function 154 General remarks on analytic functions 156 Preliminary discussion of singular points 157 Transcendental integral functions 159 Natural boundaries 160 The meaning of a 168 Mapping with the circular functions 175 Nonessential singular points 181 Transcendental fractional functions 187 CHAPTER XV 194 Formulae for the other circular functions 204 Reconciliation of the definitions in the case of the power 211 ART PAGE 117 Case where the endvalues belong to different elements 214 Cauchys theorem 218 Residues 219 General applications of the theory of residues 222 Special applications to real definite integrals 223 CHAPTER XVII 230 Isolated singularities of onevalued functions 232 Fouriers series 235 The partitionfunction 237 The theta functions 240 CHAPTER XVIII 243 A theorem on convergence 245 The functions an Jf 246 Series for J au in powers of u 249 Double periodicity 250 The zeros of fpu 251 Are ip i a periodic? 252 CHAPTER XIX 255 Comparison of elliptic functions 258 Algebraic equation connecting the functions u ffftt 259 The addition theorem for j 260 Expression of an elliptic function by means of u 262 The addition theorem for f 263 Integration of an elliptic function 264 Expression of an elliptic function by means of au 265 Relation connecting J au 268 The function Ja 270 CHAPTER XX 273 Corresponding paths in the x _yplanes when yix 277 Example 2 y2xax6 280 Example 3 Rational functions of xy wherey2xajxb 282 Example 4 y3y 2x 284 Simply connected Riemann surface 286 Example 5 yixa1 xaxa3xa 287 Fundamental regions 289 CHAPTER XXI 293 Proof that an algebraic function is analytic 294 Puiseux series 296 Double points on the curve Fxyo 298 Infinite values of the variables 299 The singular points of an algebraic function 301 An algebraic equation in x y defines a single function 302 Riemann surface for an algebraic function 304 CHAPTER XXII 306 Difficulties underlying Cauchys definition 311 Extended form of Taylors theorem 313 The potential 315 The equipotential problem 317 Schwarzs and Christoffels mapping of a straight line on a polygon 321 Greens theorem for two dimensions 322 Cauchys theorem 324 List of books 327

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JSTOR: Introduction to the Theory of Analytic Functions
Introduction to the Theory of Analytic Functions. By PROFESSORS J. HARKNESS, ma, and F. MORLEY, Sc.D. (Macmillan and Co, 8vo., pp. xv. +366. ...

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Frank Morley - Wikipedia, the free encyclopedia
(1893), with James Harkness; and Introduction to the Theory of Analytic Functions (1898).[1] He was President of the American Mathematical Society from 1919 ...
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Historical Math Monographs
Title: Introduction to the theory of analytic functions. Publication Info: Ithaca, New York: Cornell University Library ...
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Earliest Known Uses of Some of the Words of Mathematics (C)
Circle of convergence also appears in 1898 in Introduction to the theory of analytic functions by Harkness and Morley: "Hence there is a frontier value R ...
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