Analysis III: Analytic and Differential Functions, Manifolds and Riemann Surfaces

325 Pages·2015·2.18 MB·English

by  Roger Godement

Universitext
Roger Godement
Analysis III
Analytic and Differential Functions,
Manifolds and Riemann Surfaces
Universitext
Universitext
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Roger Godement
Analysis III
Analytic and Differential Functions,
Manifolds and Riemann Surfaces
Translated by Urmie Ray
Roger Godement
Paris, France
Translated by Urmie Ray

Translation from the French language edition: Analyse mathematique III by Roger Godement,
Copyright © Springer-Verlag GmbH Berlin Heidelberg 2002.
Springer-Verlag GmbH Berlin Heidelberg is part of Springer Science+Business Media

ISSN 0172-5939 ISSN219 1-6675 (ele ctronic)
Universitext
ISBN 978-3-319-16052-8 ISBN978 -3-319-16053-5 (eB ook)
DOI 10.1007/978-3-319-16053-5

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Table of Contents of Volume III
VIII – Cauchy Theory ........................................ 1
§1. Integrals of Holomorphic Functions .......................... 3
1 – Preliminary Results ..................................... 3
2 – The Problem of Primitives ............................... 8
3 – Homotopy Invariance of Integrals ......................... 18
§2. Cauchy’s Integral Formulas ................................. 31
4 – Integral Formula for a Circle.............................. 31
5 – The Residue Formula.................................... 42
6 – Dixon’s Theorem........................................ 56
7 – Integrals Depending Holomorphically on a Parameter........ 59
§3. Some Applications of Cauchy’s Method ...................... 63
8 – Fourier Transform of a Rational Fraction................... 65
9 – Summation Formulas .................................... 75
10 –The Gamma Function, The Fourier Transform of e−xxs−1
+
and the Hankel Integral .................................. 78
11 – The Dirichlet problem for the half-plane .................. 86
12 – The Complex Fourier Transform ......................... 94
13 – The Mellin Transform .................................. 103
14 – Stirling’s Formula for the Gamma Function................ 118
15 – The Fourier Transform of 1/cosh πx...................... 125
IX – Multivariate Differential and Integral Calculus .......... 133
§1. Classical Differential Calculus ............................... 133
1 – Linear Algebra and Tensors .............................. 133
(i) Finite-dimensional vector spaces ........................ 133
(ii) Tensor notation...................................... 135
2 – Differential Calculus of n Variables ........................ 146
(i) Differential functions.................................. 146
(ii) Multivariate chain rule ............................... 149
(iii) Partial differentials .................................. 151
(iv) Diffeomorphisms .................................... 154
(v) Immersions, submersions, subimmersions ............... 155
3 – Calculations in Local Coordinates ......................... 155
V
VI Table of Contents of Volume III
(i) Diffeomorphisms and local charts ....................... 155
(ii) Moving frames and tensor fields........................ 157
(iii) Covariant derivatives on a Cartesian space ............. 161
§2. Differential Forms of Degree 1............................... 166
4 – Differential Forms of Degree 1 ............................ 166
5 – Local Primitives ........................................ 168
(i) Existence: calculations in terms of coordinates ........... 168
(ii) Existence of local primitives: intrinsic formulas .......... 170
6 – Integration Along a Path. Inverse Images................... 172
(i) Integrals of a differential form.......................... 172
(ii) Inverse image of a differential form .................... 173
7 – Effect of a Homotopy on an Integral ....................... 175
(i) Differentiation with respect to a path .................... 175
(ii) Effect of a homotopy on an integral .................... 177
(iii) The Banach space C1/2(I;E) ......................... 179
§3. Integration of Differential Forms............................. 182
8 – Exterior Derivative of a Form of Degree 1 .................. 182
(i) Physicists’ Vector Analysis............................. 182
(ii) Differential forms of degree 2 .......................... 183
(iii) Forms of degree p.................................... 186
9 – Extended Integrals over a 2-Dimensional Path .............. 191
(i) The exterior derivative as an infinitesimal integral........ 192
(ii) Stokes’ formula for a 2-dimensional path ................ 195
(iii) Integral of an inverse image........................... 197
(iv) A planar example .................................... 199
(v) Classical version ..................................... 200
10 – Change of Variables in a Multiple Integral................. 203
(i) Case where ϕ is linear................................. 204
(ii) Approximation Lemmas............................... 208
(iii) Change of variable formula ........................... 213
(iv) Stokes’ formula for a p-dimensional path................ 215
§4. Differential Manifolds ...................................... 218
11 – What is a Manifold? ................................... 218
(i) The sphere in R3 ..................................... 218
(ii) The notion of a manifold of class Cr and dimension d .... 219
(iii) Some Examples ..................................... 221
(iv) Differentiable maps .................................. 224
12 – Tangent vectors and Differentials......................... 225
(i) Vectors and tangent vector spaces ...................... 225
(ii) Tangent vector to a curve ............................. 228
(iii) Differential of a map................................. 230
(iv) Partial differentials .................................. 233
(v) The manifold of tangent vectors ........................ 233
13 – Submanifolds and Subimmersions ........................ 235
(i) Submanifolds ........................................ 236
Table of Contents of Volume III VII
(ii) Submanifolds defined by a subimmersion ................ 239
(iii) One-Parameter Subgroups of a Torus .................. 242
(iv) Submanifolds of a Cartesian space: tangent vectors ...... 246
(v) Riemann spaces...................................... 248
14 – Vector Fields and Differential Operators .................. 250
15 – Vector Fields and Differential Equations .................. 252
(i) Reduction to an integral equation ....................... 253
(ii) Existence of solutions................................. 254
(iii) Uniqueness of the solution ............................ 255
(iv) Dependence on initial conditions....................... 255
(v) Matrix exponential . .................................. 258
16 – Differential Forms on a Manifold ......................... 261
17 – Integration of Differentiable Forms ....................... 262
(i) Orientable manifolds .................................. 263
(ii) Integration of differential forms ........................ 266
18 – Stokes’ Formula........................................ 269
X – The Riemann Surface of an Algebraic Function........... 275
1 – Riemann Surfaces ....................................... 275
2 – Algebraic Functions ..................................... 280
3 – Coverings of a Topological Space.......................... 285
(i) Definition of a covering ............................... 285
(ii) Sections of a covering space ........................... 287
(iii) Path-lifting ......................................... 288
(iv) Coverings of a simply connected space .................. 292
(v) Coverings of a pointed disc ............................ 295
4 – The Riemann Surface of an Algebraic Function ............. 297
(i) Global uniform branches ............................... 297
(ii) Definition of a Riemann surface Xˆ ..................... 298
(iii) The algebraic function F(z) as a meromorphic function
on Xˆ ............................................... 300
(iv) Connectedness of Xˆ ................................. 303
(v) Meromorphic functions on Xˆ .......................... 305
(vi) The purely algebraic point of view ..................... 306
Index.........................................................311
Table of Contents of Volume I ................................315
Table of Contents of Volume II ...............................319
VIII – Cauchy Theory
§1. Integrals of holomorphic functions – §2. Cauchy’s Integral
Formulas – §3. Some Applications of Cauchy’s Method
InChapterVII,§4weshowedhowasignificantpartoftheclassicaltheory
ofholomorphicoranalyticfunctionsonCcanbeobtainedfromFourierseries.
In fact, our universal method for constructing them – a fundamental idea
of Cauchy – is to integrate holomorphic functions along curves drawn in
theirdomainsofdefinitionandtherebyobtainaversionofthe“fundamental
theoremofdifferentialandintegralcalculus”(FT)forholomorphicfunctions,
and then to deduce countless consequences.
I will present only very few of them. The general theory of analytic func-
tions is of unlimited scope1 and the results needed in mathematical fields
where holomorphic functions are encountered are, on the other hand, very
limited in most cases. For instance, a famous result like Riemann’s theorem
onconformalmappingofsimplyconnecteddomainsisrarelyused,althoughit
is recommended to be familiar with it for the sake of “general knowledge”;
as for classifying simply connected Riemann surfaces, which would be far
more useful, it would need far too complicated developments. The very basic
results and methods that we will present in this chapter are, for example,
quite sufficient for the chapter devoted to the theory of Riemann surfaces or
for the one on elliptic and modular functions.
1 ThetwovolumesbyReinholdRemmert,Funktionentheorie (Springer,1995,also
available in English edition), more than 700 very compact pages, can give some
idea of the general theory of analytic functions, but do not cover Riemann sur-
faces, elliptic and automorphic functions, differential equations in the complex
domain,specialfunctions,etc.,areasthatwouldrequirethousandsofadditional
pages and that, at any rate, have been the subject of specialized presentations.
OthernumerousavailablepresentationsincludeWalterRudin,RealandComplex
Analysis (McGraw-Hill, 1966, also available in French), Jean Dieudonn´e, Calcul
Infinit´esimal (Hermann,1968),inparticularusefulforitsmanyexercises,Eber-
hard Freitag & Rolf Busam, Funktionentheorie (Springer-Verlag, 1995), which
lists several other books, Serge Lang, Complex Analysis (Springer, many edi-
tions), John B. Conway, Functions of One Complex Variable (2 vol., Springer,
1978–95),CarlosA.Berenstein&RogerGray,Complex Variables. An Introduc-
tion (Springer, 1991).
© Springer International Publishing Switzerland 2015 1
R. Godement, Analysis III, Universitext, DOI 10.1007/978-3-319-16053-5_1
2 VIII – Cauchy Theory
Learning to use basic ideas is, therefore, much better than learning a
multitude of general theorems, albeit ingenious and deep, unless, of course,
the aim is to specialize in general theory.
Cauchytheory(itwouldbemuchbettertosay:CauchyandWeierstrass)
has been and continues to be the subject of many accounts merely differing
from each other on detailed points of presentation or style; not seeing the
need to reproduce them an umpteenth time, I have tried, whenever possi-
ble, not to follow them, in particular regarding homotopy. As will be seen
in the next chapter, apart from the residue theorem, another easy conse-
quence, Cauchy’s method falls within the much more general framework of
multivariate differential forms.