Holomorphic Functions in the Plane and n-dimensional SpaceComplex analysis nowadays has higher-dimensional analoga: the algebra of complex numbers is replaced then by the non-commutative algebra of real quaternions or by Clifford algebras. During the last 30 years the so-called quaternionic and Clifford or hypercomplex analysis successfully developed to a powerful theory with many applications in analysis, engineering and mathematical physics. This textbook introduces both to classical and higher-dimensional results based on a uniform notion of holomorphy. Historical remarks, lots of examples, figures and exercises accompany each chapter. |
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Holomorphic Functions in the Plane and n-dimensional Space Klaus Gürlebeck,Klaus Habetha,Wolfgang Sprößig No preview available - 2009 |
Common terms and phrases
analogous arbitrary automorphic basis elements boundary calculated called Cauchy Cauchy–Riemann Cauchy’s integral formula Cl(n Clifford algebra coefficients complex numbers Corollary corresponding cosh curve defined Definition denoted derivatives differential equations differential form disc disc of convergence domain G Eisenstein series Example Exercise exists exponential function exterior f is holomorphic finite follows Fueter polynomials function f geometric given H-holomorphic Hölder holds holomorphic function integral theorem Laurent series left-holomorphic Let f Let G linear manifold mapping matrix meromorphic Möbius transformations multiplication obtain operator orthogonal paravector piecewise smooth plane poles power series Proof properties Proposition prove quaternions radius real number Remark removable singularity representation residue resp Riemann Riemann sphere Rn+1 rotation scalar product sequence singularity sinh sphere spherical functions straight line subset subspaces Taylor series variables vector product vector space zero Zeta function