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Quaternionic analysis

  • A. Sudbery (a1)
Abstract

The richness of the theory of functions over the complex field makes it natural to look for a similar theory for the only other non-trivial real associative division algebra, namely the quaternions. Such a theory exists and is quite far-reaching, yet it seems to be little known. It was not developed until nearly a century after Hamilton's discovery of quaternions. Hamilton himself (1) and his principal followers and expositors, Tait(2) and Joly(3), only developed the theory of functions of a quaternion variable as far as it could be taken by the general methods of the theory of functions of several real variables (the basic ideas of which appeared in their modern form for the first time in Hamilton's work on quaternions). They did not delimit a special class of regular functions among quaternion-valued functions of a quaternion variable, analogous to the regular functions of a complex variable.

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(4)R. Fueter Die Funktionentheorie der Differentialgleichungen Δu = 0 und ΔΔu = 0 mit vier reellen Variablen. Comment. Math. Helv. 7 (1935), 307330.

(5)R. Fueter Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen. Comment. Math. Helv. 8 (1936), 371378.

(6)H. Haefeli Hyperkomplexe Differentiate. Comment. Math. Helv. 20 (1947), 382420.

(7)C. A. Deavours The quaternion calculus. Amer. Math. Monthly 80 (1973), 9951008.

(8)B. Schuler Zur Theorie der regulären Funktionen einer Quaternionen-Variablen. Comment. Math. Helv. 10 (1937), 327342.

(9)R. Fueter Die Singularitäten der eindeutigen regulären Funktionen einer Quaternionen-variablen. Comment. Math. Helv. 9 (1937), 320335.

(15)C. G. Cullen An integral theorem for analytic intrinsic functions on quaternions. Duke Math. J. 32 (1965), 139148.

(16)J. A. Schouten Ricci-calculus, 2nd ed. (Berlin, Springer-Verlag, 1954).

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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