Quaternionic analysis: Difference between revisions

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The success of [[complex analysis]] in providing a rich family of [[holomorphic function]]s for scientific work has engaged some workers in efforts to extend the planar theory, based on complex numbers, to a 4-space study with functions of a quaternion variable.<ref>{{harv|Fueter|1936}}</ref> These efforts were summarized in {{harvtxt|Deavours|1973}}.{{efn|{{harvtxt|Deavours|1973}} recalls a 1935 issue of ''[[Commentarii Mathematici Helvetici]]'' where an alternative theory of "regular functions" was initiated by {{harvtxt|Fueter|1936}} through the idea of [[Morera's theorem]]: quaternion function <math>F</math> is "left regular at <math>q</math>" when the integral of <math>F</math> vanishes over any sufficiently small [[hypersurface]] containing <math>q</math>. Then the analogue of [[Liouville's theorem (complex analysis)|Liouville's theorem]] holds: The only regular quaternion function with bounded norm in <math>\mathbb{E}^4</math> is a constant. One approach to construct regular functions is to use [[power series]] with real coefficients. Deavours also gives analogues for the [[Poisson integral]], the [[Cauchy integral formula]], and the presentation of [[Maxwell’s equations]] of electromagnetism with quaternion functions.}}

Though <math>\mathbb{H}</math> [[quaternion#H asAs a union of complex planes|appears as a union of complex planes]], the following proposition shows that extending complex functions requires special care:

Let <math>f_5(z) = u(x,y) + i v(x,y)</math> be a function of a complex variable, <math>z = x + i y</math>. Suppose also that <math>u</math> is an [[even function]] of <math>y</math> and that <math>v</math> is an [[odd function]] of <math>y</math>. Then <math>f_5(q) = u(x,y) + rv(x,y)</math> is an extension of <math>f_5</math> to a quaternion variable <math>q = x + yr</math> where <math>r^2 = -1</math> and <math>r \in \mathbb{H}</math>.