Quaternionic analysis
1979sud | Sudbery Antony

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 and his principal followers and expositors, Tait and Joly, 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.
This may have been because neither of the two fundamental definitions of a regular function of a complex variable has interesting consequences when adapted to quaternions; one is too restrictive, the other not restrictive enough. The functions of a quaternion variable which have quaternionic derivatives, in the obvious sense, are just the constant and linear functions (and not all of them); the functions which can be represented by quaternionic power series are just those which can be represented by power series in four real variables.
In 1935 R Fueter proposed a deашnition of "regular" for quaternionic functions by means of an analogue of the Cauchy-Riemann equations. He showed that this definition led to close analogues of Cauchy's theorem Cauchy's integral formula, and the Laurent expansion. In the next twelve years Fueter and his collaborators developed the theory of quaternionic analysis.
The theory developed by Fueter and his school is incomplete in some ways, and many of their theorems are neither so general nor so rigorously proved as present-day standards of exposition in complex analysis would require. The purpose of this paper is to give a self-contained account of the main line of quaternionic analysis which remedies these deficiencies, as well as adding a certain number of new results. By using the exterior differential calculus we are able to give new and simple proofs of most of the main theorems and to clarify the relationship between quaternionic analysis and complex analysis.

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