Abstract
The zero rest mass Euclidean Dirac equations in 2 (4) dimensions may be regarded as ‘square roots’ of the second order harmonic equation, and give rise to the crucial integral theorem and integral formula of complex (quaternionic) analysis. Recently discovered ‘2rth root’ equations for the 2rth order ‘harmonic’ equations are here shown to give rise to a similar integral theorem and integral formula.
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The integral theorem and formula were discovered by R. Fueter inComment. Math. Helv. 7, 307 (1935) and8, 371 (1936), and developed by Fueter and collaborators (who also apparently considered certain Clifford algebras other thanH) for several years (see H. Haefeli inComment. Math. Helv. 20, 382 (1947)).
My interest in quaternionic analysis started with a rediscovery of the elementary aspects of the theory in C. J. S., Clarke, P. J. McCarthy, and A. Sudbery, ‘A Quaternion Analogue of Complex Analysis’ (unpublished manuscript, September 1976), which derived the integral theorem and formula using a method similar to the one used above. An elementary account (using different methods) of the key theorems was given by C. A. Deavours inAmer. Math. Monthly 80, 995 (1973). A. Sudbery has now written a review which incorporates some of the earlier work, derives the key theorems in a general way, and gives some additional results.Math. Proc. Camb. Phil. Soc. 85, No. 2199 (1979).
McCarthyP.J.,Lett. Math. Phys. 4, 39 (1980).
In the casesr=2, arbitraryn≥2, the Green's functions are well known;\(G_q (x) = \left\{ \begin{gathered} (2\pi )^{ - 1} \log \rho (n = 2) \hfill \\ [(2 - n)o_n ]^{ - 1} \rho ^{2 - n} (n > 2), \hfill \\ \end{gathered} \right.\) where ρ=∣x−q∣ and σ 2 is the surface area of the unit sphereS n−1 inn dimensions. Even in these cases, some of the results above seem to be new. For example, the three-dimensional Laplace equationφ xx +φ yy +φ zz =0 has a ‘generalised complex analysis’ based on the analyticity equation\(\vec \partial f = if_x + if_y + kf_z = 0{\text{ (}}f{\text{: R}}^{\text{3}} \to H).\)Here the KernelK q , given above, is completely explicit.
This is to be expected, sinceD is no longer rotationally invariant. It is translationally invariant, however, so that\(G_q (x) = {\text{ }}G_{\text{o}} (x - q).\)
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McCarthy, P.J. Ultrageneralised complex analysis. Lett Math Phys 4, 509–514 (1980). https://doi.org/10.1007/BF00943438
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DOI: https://doi.org/10.1007/BF00943438