Quaternionic analysis: Difference between revisions

Revision as of 10:53, 15 April 2018

Revision as of 17:31, 31 May 2018

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As with complex and real analysis, it is possible to study the concepts of [[analytic function|analyticity]], [[holomorphic function|holomorphy]], [[harmonic function|harmonicity]] and [[conformality]] in the context of quaternions. It is known that for the complex numbers, these four notions coincide; however, for the quaternions, and also the real numbers, not all of the notions are the same.

==DiscussionProperties==

The [[projection (linear algebra)|projections]] of a quaternion onto its scalar part or onto its vector part, as well as the modulus and [[versor]] functions, are examples that are basic to understanding quaternion structure.

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Quaternion variable theory differs in some respects from complex variable theory. For example: The [[complex conjugate]] mapping of the complex plane is a central tool but requires the introduction of a non-arithmetic, [[Holomorphic function|non-analytic]] operation. Indeed, conjugation changes the [[orientation (mathematics)|orientation]] of plane figures, something that arithmetic functions do not change.

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 can be proven by taking the basis elements. With them, we have

'''Proposition:''' The function <math>f_4(q) = - \tfrac{1}{2} (q + iqi + jqj + kqk)</math> is equivalent to quaternion conjugation.

'''Proof''': For the basis elements we have

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

'''Proposition:''' 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>.

'''Proof''':Then, Letlet <math>r^*</math> represent the conjugate of <math>r</math>, so that <math>q = x - yr^*</math>. The extension to <math>\mathbb{H}</math> will be complete when it is shown that <math>f_5(q) = f_5(x - yr^*)</math>. Indeed, by hypothesis

:<math>u(x,y) = u(x,-y), \quad v(x,y) = -v(x,-y) \quad</math> so that one obtains

'''Proof''': Let <math>r^*</math> represent the conjugate of <math>r</math>, so that <math>q = x - yr^*</math>. The extension to <math>\mathbb{H}</math> will be complete when it is shown that <math>f_5(q) = f_5(x - yr^*)</math>. Indeed, by hypothesis

:<math>u(x,y) = u(x,-y), \quad v(x,y) = -v(x,-y) \quad</math> so that one obtains

:<math>f_5(x - y r^*) = u(x,-y) + r^* v(x,-y) = u(x,y) + r v(x,y) = f_5(q).</math>