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By Klaus Gürlebeck

Complex research these days has higher-dimensional analoga: the algebra of complicated numbers is changed then by means of the non-commutative algebra of actual quaternions or by means of Clifford algebras. over the past 30 years the so-called quaternionic and Clifford or hypercomplex research effectively constructed to a strong thought with many purposes in research, engineering and mathematical physics. This textbook introduces either to classical and higher-dimensional effects in keeping with a uniform inspiration of holomorphy. ancient comments, plenty of examples, figures and routines accompany every one chapter.

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Five functions . . . . . . . . . . . . . . . . . . 2. five. 1 Visualization of the sector S three . . . 2. five. 2 parts of round trigonometry . 2. 6 routines . . . . . . . . . . . . . . . . . . . . three Clifford numbers . . . . . . . . . . . . . . . . . . . . three. 1 historical past of the invention . . . . . . . . . . . . three. 2 Definition and houses . . . . . . . . . . . three. 2. 1 Definition of the Clifford algebra . . three. 2. 2 buildings and automorphisms . . . three. 2. three Modulus . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 2 three 10 thirteen 15 15 sixteen 24 24 26 30 31 33 37 forty two forty six forty six forty seven forty nine 50 50 fifty two fifty two fifty five fifty eight vi Contents three. three . . . . . . . . . . . . . . . . . . . . . . . . sixty one sixty one sixty three sixty six sixty seven seventy one services Topological facets . . . . . . . . . . . . . . . . . . . . . . . . four. 1 Topology and continuity . . . . . . . . . . . . . . . . . four. 2 sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . four. three Riemann spheres . . . . . . . . . . . . . . . . . . . . . four. three. 1 complicated case . . . . . . . . . . . . . . . . . . four. three. 2 greater dimensions . . . . . . . . . . . . . . . four. four routines . . . . . . . . . . . . . . . . . . . . . . . . . five Holomorphic capabilities . . . . . . . . . . . . . . . . . . . . . . five. 1 Differentiation in C . . . . . . . . . . . . . . . . . . . . five. 2 Differentiation in H . . . . . . . . . . . . . . . . . . . five. 2. 1 Mejlikhzhon’s outcome . . . . . . . . . . . . . . five. 2. 2 H-holomorphic features . . . . . . . . . . . five. 2. three Holomorphic capabilities and differential varieties five. three Differentiation in C (n) . . . . . . . . . . . . . . . . . five. four workouts . . . . . . . . . . . . . . . . . . . . . . . . . 6 Powers and Möbius transforms . . . . . . . . . . . . . . . . . 6. 1 Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. 1. 1 Powers in C . . . . . . . . . . . . . . . . . . 6. 1. 2 Powers in larger dimensions . . . . . . . . . 6. 2 Möbius differences . . . . . . . . . . . . . . . . . 6. 2. 1 Möbius changes in C . . . . . . . . . 6. 2. 2 Möbius variations in larger dimensions 6. three workouts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . seventy three seventy four seventy four seventy nine eighty three eighty three 87 88 ninety ninety ninety five ninety six ninety seven a hundred and one 104 107 108 108 108 109 114 114 118 124 III Integration and fundamental theorems 7 necessary theorems and indispensable formulae . . . . . . . . . . . . . . . 7. 1 Cauchy’s vital theorem and its inversion . . . . . . . . . 7. 2 Formulae of Borel–Pompeiu and Cauchy . . . . . . . . . . . 7. 2. 1 formulation of Borel–Pompeiu . . . . . . . . . . . . . 7. 2. 2 formulation of Cauchy . . . . . . . . . . . . . . . . . . 7. 2. three Formulae of Plemelj–Sokhotski . . . . . . . . . . . 7. 2. four historical past of Cauchy and Borel–Pompeiu formulae . 7. three effects of Cauchy’s imperative formulation . . . . . . . . . 7. three. 1 better order derivatives of holomorphic services 7. three. 2 suggest worth estate and greatest precept . . . 7. three. three Liouville’s theorem . . . . . . . . . . . . . . . . . . one hundred twenty five 126 126 129 129 131 133 138 141 141 a hundred and forty four 146 three. four three. five II four Geometric purposes . . . . . . . . . . three. three. 1 Spin teams . . . . . . . . . . . three. three. 2 development of rotations of Rn three. three. three Rotations of Rn+1 . . . . . . . Representations . . . . . . . . . . . . . . routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents eight IV 7. three. four quintessential formulae of Schwarz and Poisson . . . 7. four workouts . . . . . . . . . . . . . . . . . . . . . . . . . . Teodorescu remodel . . . . . . . . . . . . . . . . . . . . . . . eight. 1 houses of the Teodorescu remodel . . . . . . . . . eight. 2 Hodge decomposition of the quaternionic Hilbert house .

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