Introduction to the Theory of Analytic Functions (Google eBook)

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Macmillan and Company, Limited, 1898 - Functions - 336 pages
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Contents

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
Cauchys theorem on the coefficients of a power series
129
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
Index
328

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Page 87 - If in each row of a determinant the absolute value of the element on the principal diagonal is greater than the sum of the absolute values of the remaining elements in that row, the value of the determinant is different from zero. PROOF Let | A \ be an...
Page vii - ... a foundation of algebraic truths. It is therefore not correct to turn around and, expressing myself briefly, use "transcendental...
Page 325 - Also we see that, by making n sufficiently large, we can make the fraction - as small as we please. Thus by taking a sufficient number of terms the sum can be made to differ by as little as we please from 2. In the next article a more general case is discussed.
Page 67 - P and a neighbouring point on the curve can be made to differ from it by as little as we please...
Page 112 - Sc is also absolutely convergent, and its sum is the product of the sums of the two former series.
Page 14 - ... levers: the first has the fulcrum between the power and weight; in the second the weight acts between the fulcrum and the power; and in the third the power acts between the fulcrum and the weight. PROP. To find the conditions of equilibrium of two forces acting in the same plane on a lever. 93. Let the plane of the...
Page 4 - ... and not before the other. It is very important to notice that we have now a closed number-system. When we seek to separate the irrational objects as lying left or right of an object, either the object is rational, or if not it separates rational objects and is irrational ; in any case...
Page 123 - The idea that series of powers are as serviceable for algebra as for arithmetic was first worked out by Newton*, and in the theory of functions of a complex variable, as it now stands, the theory of such series is the solid foundation for the whole structure.
Page 3 - ... comes first. It is to be noticed that as we approach any of the natural objects there is no last fractional mark ; that is, whatever object we take there are always others between it and the natural object.

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