Preface to the German Edition .................................. xi
Preface to the English Edition ................................ xiv
I Numbers ..................................................... 1
1 Complex numbers .............................................. 2
1.1 The History of Their Discovery .......................... 2
1.2 Definition and Properties ............................... 3
1.3 Representations and geometric aspects .................. 10
1.4 Exercises .............................................. 13
2 Quaternions ................................................. 15
2.1 The history of their discovery ......................... 15
2.2 Definition and properties .............................. 16
2.3 Mappings and representations ........................... 24
2.3.1 Basic maps ...................................... 24
2.3.2 Rotations in  3 ................................. 26
2.3.3 Rotations of 4 ................................. 30
2.3.4 Representations ................................. 31
2.4 Vectors and geometrical aspects ........................ 33
2.4.1 Bilinear products ............................... 37
2.4.2 Multilinear products ............................ 42
2.5 Applications ........................................... 46
2.5.1 Visualization of the sphere S3 .................. 46
2.5.2 Elements of spherical trigonometry .............. 47
2.6 Exercises .............................................. 49
3 Clifford numbers ............................................ 50
3.1 History of the discovery ............................... 50
3.2 Definition and properties .............................. 52
3.2.1 Definition of the Clifford algebra .............. 52
3.2.2 Structures and automorphisms .................... 55
3.2.3 Modulus ......................................... 58
3.3 Geometrie applications ................................. 61
3.3.1 Spin groups ..................................... 61
3.3.2 Construction of rotations of n ................. 63
3.3.3 Rotations of n+1 ............................... 66
3.4 Representations ........................................ 67
3.5 Exercises .............................................. 71
II Functions .................................................. 73
4 Topological aspects ......................................... 74
4.1 Topology and continuity ................................ 74
4.2 Series ................................................. 79
4.3 Riemann spheres ........................................ 83
4.3.1 Complex case .................................... 83
4.3.2 Higher dimensions ............................... 87
4.4 Exercises .............................................. 88
5 Holomorphic functions ....................................... 90
5.1 Differentiation in  ................................... 90
5.2 Differentiation in  ................................... 95
5.2.1 Mejlikhzhon's result ............................ 96
5.2.2 H-holomorphic functions ......................... 97
5.2.3 Holomorphic functions and differential forms ... 101
5.3 Differentiation in ℓ(n) .............................. 104
5.4 Exercises ............................................. 107
6 Powers and Mцbius transforms ............................... 108
6.1 Powers ................................................ 108
6.1.1 Powers in .................................... 108
6.1.2 Powers in higher dimensions .................... 109
6.2 Mцbius transformations ................................ 114
6.2.1 Mцbius transformations in .................... 114
6.2.2 Mцbius transformations in higher dimensions .... 118
6.3 Exercises ............................................. 124
III Integration and integral theorems ........................ 125
7 Integral theorems and integral formulae .................... 126
7.1 Cauchy's integral theorem and its inversion ........... 126
7.2 Formulae of Borel-Pompeiu and Cauchy .................. 129
7.2.1 Formula of Borel-Pompeiu ....................... 129
7.2.2 Formula of Cauchy .............................. 131
7.2.3 Formulae of Plemelj-Sokhotski .................. 133
7.2.4 History of Cauchy and Borel-Pompeiu formulae ... 138
7.3 Consequences of Cauchy's integral formula ............. 141
7.3.1 Higher order derivatives of holomorphic
functions ...................................... 141
7.3.2 Mean value property and maximum principle ...... 144
7.3.3 Liouville's theorem ............................ 146
7.3.4 Integral formulae of Schwarz and Poisson ....... 147
7.4 Exercises ............................................. 149
8 Teodorescu transform ....................................... 151
8.1 Properties of the Teodorescu transform ................ 151
8.2 Hodge decomposition of the quaternionic Hilbert
space ................................................. 156
8.2.1 Hodge decomposition ............................ 156
8.2.2 Representation theorem ......................... 159
8.3 Exercises ............................................. 160
IV Series expansions and local behavior ....................... 161
9 Power series ............................................... 162
9.1 WeierstraЯ' convergence theorems, power series ........ 162
9.1.1 Convergence theorems according to WeierstraЯ ... 162
9.1.2 Power series in .............................. 164
9.1.3 Power series in ℓ(n) .......................... 167
9.2 Taylor and Laurent series in ........................ 169
9.2.1 Taylor series .................................. 169
9.2.2 Laurent series ................................. 173
9.3 Taylor and Laurent series in ℓ(n) .................... 175
9.3.1 Taylor series .................................. 175
9.3.2 Laurent series ................................. 181
9.4 Exercises ............................................. 184
10 Orthogonal expansions in ................................. 186
10.1 Complete H-holomorphic function systems ............... 186
10.1.1 Polynomial systems ............................. 188
10.1.2 Inner and outer spherical functions ............ 191
10.1.3 Harmonic spherical functions ................... 194
10.1.4 H-holomorphic spherical functions .............. 196
10.1.5 Completeness in L2(B3) ∩ ker  ................. 202
10.2 Fourier expansion in ................................ 203
10.3 Applications .......................................... 203
10.3.1 Derivatives of -holomorphic polynomials ....... 203
10.3.2 Primitives of -holomorphic functions .......... 207
10.3.3 Decomposition theorem and Taylor expansion ..... 213
10.4 Exercises ............................................. 215
11 Elementary functions ....................................... 218
11.1 Elementary functions in ............................. 218
11.1.1 Exponential function ........................... 218
11.1.2 Trigonometric functions ........................ 219
11.1.3 Hyperbolic functions ........................... 221
11.1.4 Logarithm ...................................... 223
11.2 Elementary functions in ℓ(n) ......................... 225
11.2.1 Polar decomposition of the Cauchy-Riemann
operator ....................................... 225
11.2.2 Elementary radial functions .................... 229
11.2.3 Fueter-Sce construction of holomorphic
functions ...................................... 234
11.2.4 Cauchy-Kovalevsky extension .................... 239
11.2.5 Separation of variables ........................ 244
11.3 Exercises ............................................. 249
12 Local structure of holomorphic functions ................... 252
12.1 Behavior at zeros ..................................... 252
12.1.1 Zeros in ..................................... 252
12.1.2 Zeros in ℓ(n) ................................. 255
12.2 Isolated singularities of holomorphic functions ....... 259
12.2.1 Isolated singularities in .................... 259
12.2.2 Isolated singularities in ℓ(n) ................ 265
12.3 Residue theorem and the argument principle ............ 267
12.3.1 Residue theorem in ........................... 267
12.3.2 Argument principle in ........................ 270
12.3.3 Residue theorem in ℓ(n) ....................... 274
12.3.4 Argument principle in ℓ(n) .................... 276
12.4 Calculation of real integrals ......................... 279
12.5 Exercises ............................................. 285
13 Special functions .......................................... 287
13.1 Euler's Gamma function ................................ 287
13.1.1 Definition and functional equation ............. 287
13.1.2 Stirling's theorem ............................. 291
13.2 Riemann's Zeta function ............................... 296
13.2.1 Dirichlet series ............................... 296
13.2.2 Riemann's Zeta function ........................ 298
13.3 Automorphic forms and functions ....................... 302
13.3.1 Automorphic forms and functions in ........... 302
13.3.2 Automorphic functions and forms in ℓ(n) ....... 307
13.4 Exercises ............................................. 321
Appendix ...................................................... 323
A.l Differential forms in n .................................. 324
A.1.1 Alternating linear mappings ......................... 324
A.1.2 Differential forms .................................. 329
A.1.3 Exercises ........................................... 336
A.2 Integration and manifolds ................................. 338
A.2.1 Integration ......................................... 338
A.2.1.1 Integration in n+1 ........................ 338
A.2.1.2 Transformation of variables ................ 339
A.2.1.3 Manifolds and integration .................. 341
A.2.2 Theorems of Stokes, Gauß, and Green ................. 351
A.2.2.1 Theorem of Stokes .......................... 351
A.2.2.2 Theorem of Gauß ............................ 352
A.2.2.3 Theorem of Green ........................... 354
A.2.3 Exercises ........................................... 355
A.3 Some function spaces ...................................... 357
A.3.1 Spaces of Hölder continuous functions ............... 357
A.3.2 Spaces of differentiable functions .................. 358
A.3.3 Spaces of integrable functions ...................... 359
A.3.4 Distributions ....................................... 360
A.3.5 Hardy spaces ........................................ 361
A.3.6 Sobolev spaces ...................................... 361
A.4 Properties of holomorphic spherical functions ............. 363
A.4.1 Properties of Legendre polynomials .................. 363
A.4.2 Norm of holomorphic spherical functions ............. 364
A.4.3 Scalar products of holomorphic spherical functions .. 368
A.4.4 Complete orthonormal systems in  +n,H ............... 370
A.4.5 Derivatives of holomorphic spherical functions ...... 374
A.4.6 Exercises ........................................... 375
Bibliography .................................................. 377
Index ......................................................... 385
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