Holomorphic functions in the plane and n-dimensional space

Holomorphic functions in the plane and n-dimensional space

by Klaus Gürlebeck, Klaus Habetha, Wolfgang Sprößig

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About
Complex 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.

Discussion questions for Holomorphic functions in the plane and n-dimensional space

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  1. 1

    How does the transition from traditional, commutative complex analysis to the non-commutative algebra of quaternions and Clifford algebras fundamentally alter your intuition about mathematical "holomorphy"?

  2. 2

    Reflecting on your own problem-solving journey, how do you handle stepping into mathematical spaces where familiar rules—like the commutativity of multiplication—no longer apply?

  3. 3

    The authors emphasize a "uniform notion of holomorphy" across dimensions; in what ways do you think unifying disparate mathematical concepts reflects a broader human desire for symmetry and connection?

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