Quaternionic analysis: Difference between revisions

Revision as of 23:21, 8 September 2018

Revision as of 23:31, 8 September 2018

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In contrast to the [[complex conjugate]], the quaternion conjugation can be expressed arithmetically, as <math>f_4(q) = - \tfrac{1}{2} (q + iqi + jqj + kqk)</math>

This equation can be proven, bystarting takingwith the [[basis elements.(linear Withalgebra)|basis]] them{1, wei, j, havek}:

:<math>f_4(1) = -\tfrac{1}{2}(1 - 1 - 1 - 1) = 1, \quad

f_4(i) = -\tfrac{1}{2}(i - i + i + i) = -i, \quad

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Consequently, since <math>f_4</math> is [[linear function|linear]],

:<math>f_4(q) = f_4(w + x i + y j + z k) = w f_4(1) + x f_4(i) + y f_4(j) + z f_4(k) = w - x i - y j - z k = q^*.</math>

An immediate '''corollary''' of which is that the quaternion conjugate is [[Analytic function|analytic]] everywhere in <math>\mathbb{H}.</math> Compare this to the seemingly identical complex conjugate, <math>(x + iy)^* = x - iy,</math> for <math>x, y \in \mathbb{R},</math> and <math>i^2 = -1,</math> which is not [[Holomorphic function|analytic]] in <math>\mathbb{C}</math>.

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.}}