Numbers 1
1 Complex numbers ........................... 2
1.1 The History of Their Discovery . ............... 2
1.2 Definition and Properties . .................. 3
1.3 Rvpresentations 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 R3 . .................. 26
2.3.3 Rotations of R4 . .................. 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 Geometric applications ................... .. 61
3.3.1 Spin groups ....... ............. 61
3.3.2 Construction of rotations of Rn . .......... 63
3.3.3 Rotations of R"+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 C ................... ... . 90
5.2 Differentiation in HE . ................ .... 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 C?n) . ................... 104
5.4 Exercises ................... ......... 107
6 Powers and M6bius transforms ................... . 108
6.1 Powers.... ........ .... ........... 108
6.1.1 Powers in C ........ ............ 108
6.1.2 Powers in higher dimensions . ........... 109
6.2 M8bius transformations ... . . . . . . . . . . . . . 114
6.2.1 M8bius transformations in C . ........... 114
6.2.2 M6bius 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 WeierstraV' convergence theorems, power series ....... 162
9.1.1 Convergence theorems according to Weierstrafl .. 162
9.1.2 Power series in C . .................. 164
9.1.3 Power series in C (n) . ............... 167
9.2 Taylor and Laurent series in C . ............... 169
9.2.1 Taylor series . .................... 169
9.2.2 Laurent series .................... 173
9.3 Taylor and Laurent series in CU(n) . ............. 175
9.3.1 Taylor series . .................... 175
9.3.2 Laurent series . ................... 181
9.4 Exercises ................... ......... 184
10 Orthogonal expansions in HI . ................... ... 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) n ker D ... ....... . 202
10.2 Fourier expansion in H . .................. .. 203
10.3 Applications .......................... 203
10.3.1 Derivatives of H[-holomorphic polynomials ..... 203
10.3.2 Primitives of H-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 C . .................. 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 Ce(n) . ............... 225
11.2.1 Polar decomposition of the Cauchy-Riemann
operator ................... .... 225
1 1 I 0
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 C . ..................... 252
12.1.2 Zeros in C(n) . .................... 255
12.2 Isolated singularities of holomorphic functions ........ 259
12.2.1 Isolated singularities in C . ............. 259
12.2.2 Isolated singularities in Ce(n) . .......... 265
12.3 Residue theorem and the argument principle . ....... 267
12.3.1 Residue theorem in C ............. . .. 267
12.3.2 Argument principle in C . . ............. 270
12.3.3 Residue theorem in Ci(n) .............. 274
12.3.4 Argument principle in CU(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 C ...... 302
13.3.2 Automorphic functions and forms in CU(n) . . . . 307
13.4 Exercises . ........................... 321
Appendix 323
A.1 Differential forms in Rn . ....................... 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 Rn+1 .. . . . . . . . . . . . .... 338
A.2.1.2 Transformation of variables . ............ 339
A.2.1.3 Manifolds and integration . ............. 341
A.2.2 Theorems of Stokes, Gaug, and Green . ........... 351
A.2.2.1 Theorem of Stokes . ................. 351
A.2.2.2 Theorem of Gaug . ................. 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 Hl61der 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 7, . . . ........... . . 370
A.4.5 Derivatives of holomorphic spherical functions ........ 374
A.4.6 Exercises ............................ 375
Bibliography 377