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Quaternionic analysis

Published online by Cambridge University Press:  24 October 2008

A. Sudbery
Affiliation:
University of York

Extract

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.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

REFERENCES

(1)Hamilton, W. R.Elements of quaternions (London, Longmans Green, 1866).Google Scholar
(2)Tait, P. G.An elementary treatise on quaternions (Cambridge University Press, 1867).Google Scholar
(3)Joly, C. J.A manual of quaternions (London, Macmillan, 1905).Google Scholar
(4)Fueter, R.Die Funktionentheorie der Differentialgleichungen Δu = 0 und ΔΔu = 0 mit vier reellen Variablen. Comment. Math. Helv. 7 (1935), 307330.Google Scholar
(5)Fueter, R.Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen. Comment. Math. Helv. 8 (1936), 371378.CrossRefGoogle Scholar
(6)Haefeli, H.Hyperkomplexe Differentiate. Comment. Math. Helv. 20 (1947), 382420.CrossRefGoogle Scholar
(7)Deavours, C. A.The quaternion calculus. Amer. Math. Monthly 80 (1973), 9951008.CrossRefGoogle Scholar
(8)Schuler, B.Zur Theorie der regulären Funktionen einer Quaternionen-Variablen. Comment. Math. Helv. 10 (1937), 327342.CrossRefGoogle Scholar
(9)Fueter, R.Die Singularitäten der eindeutigen regulären Funktionen einer Quaternionen-variablen. Comment. Math. Helv. 9 (1937), 320335.CrossRefGoogle Scholar
(10)Porteous, I. R.Topological geometry (London, Van Nostrand Reinhold, 1969).Google Scholar
(11)Cartan, H.Elementary theory of analytic functions of one or several complex variables (London, Addison-Wesley, 1963).Google Scholar
(12)Hervé, M.Several complex variables (Oxford University Press, 1963).Google Scholar
(13)Heins, M.Complex function theory (London, Academic Press, 1968).Google Scholar
(14)Titchmarsh, E. C.The theory of functions, 2nd ed. (Oxford University Press, 1939).Google Scholar
(15)Cullen, C. G.An integral theorem for analytic intrinsic functions on quaternions. Duke Math. J. 32 (1965), 139148.Google Scholar
(16)Schouten, J. A.Ricci-calculus, 2nd ed. (Berlin, Springer-Verlag, 1954).CrossRefGoogle Scholar
(17)Sugiura, M.Unitary representations and harmonic analysis (London, Wiley, 1975).Google Scholar
(18)Mccarthy, P. J. and Sudbery, A.Harmonic analysis of generalised vector functions, generalised spin-weighted functions and induced representations. J. Phys. A 10 (1977), 331338.Google Scholar