## Advances in Analysis and Geometry: New Developments Using Clifford AlgebrasOn the 16th of October 1843, Sir William R. Hamilton made the discovery of the quaternion algebra H = qo + qli + q2j + q3k whereby the product is determined by the defining relations ·2 ·2 1 Z =] = - , ij = -ji = k. In fact he was inspired by the beautiful geometric model of the complex numbers in which rotations are represented by simple multiplications z ----t az. His goal was to obtain an algebra structure for three dimensional visual space with in particular the possibility of representing all spatial rotations by algebra multiplications and since 1835 he started looking for generalized complex numbers (hypercomplex numbers) of the form a + bi + cj. It hence took him a long time to accept that a fourth dimension was necessary and that commutativity couldn't be kept and he wondered about a possible real life meaning of this fourth dimension which he identified with the scalar part qo as opposed to the vector part ql i + q2j + q3k which represents a point in space. |

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### Contents

III | 3 |

IV | 31 |

V | 51 |

VI | 65 |

VII | 97 |

VIII | 113 |

IX | 131 |

X | 149 |

XIV | 209 |

XV | 227 |

XVI | 257 |

XVII | 287 |

XVIII | 289 |

XIX | 301 |

XX | 311 |

XXI | 345 |

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Advances in Analysis and Geometry: New Developments Using Clifford Algebras Tao Qian No preview available - 2004 |

### Common terms and phrases

adjoint Birkhauser Verlag bivectors Bloch space boundary value bracket algebra Brackx bundle Cauchy kernel classical Clifford algebra Clifford analysis Clifford bundle coefficients compact Complex Variables conformally flat conformally flat manifolds consider construction coordinates defined Definition Delanghe denote derivative Dirac operator Dirichlet distributions dv A dv Eisenstein series equation Euclidean space finite Finsler Fourier Fredholm function f geometric Giirlebeck harmonic functions harmonic spinor Hermitian higher dimensional Hilbert space holomorphic homogeneous hyperhermitian hyperholomorphic functions hypermonogenic functions hyponormal identity inequality inner product introduced invariant Laplacian lattice Lemma Let f linear Lipschitz domain Math Mathematics matrix meromorphic spinor metric Mobius transformation Monge-Ampere equation monogenic functions obtain open subset orthogonal polynomial problems Proof properties Proposition prove quaternion-valued quaternionic quaternionic analysis representation result Riemannian satisfy seminormal smooth Sommen spherical means spherical monogenics Theorem theory unit ball unit sphere vector wavelets