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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
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:
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==Homographies==
The [[quaternions and spatial rotation|rotation]] about axis ''r'' is a classical application of quaternions to [[space]] mapping.<ref>
:<math>U(q,1) \begin{pmatrix}u & 0\\0 & u \end{pmatrix} = U(qu,u) \thicksim U(u^{-1}qu, 1) ,</math>
where <math>u = \exp(\theta r) = \cos \theta + r \sin \theta</math> is a [[versor]]. If ''p'' * = −''p'', then the translation <math>q \mapsto q + p</math> is expressed by
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Such a mapping is called a [[screw displacement]]. In classical [[kinematics]], [[Chasles' theorem (kinematics)|Chasles' theorem]] states that any rigid body motion can be displayed as a screw displacement. Just as the representation of a [[Euclidean plane isometry]] as a rotation is a matter of complex number arithmetic, so Chasles' theorem, and the [[screw axis]] required, is a matter of quaternion arithmetic with homographies: Let ''s'' be a right versor, or square root of minus one, perpendicular to ''r'', with ''t'' = ''rs''.
Consider the axis passing through ''s'' and parallel to ''r''. Rotation about it is expressed<ref>{{harv|Hamilton
:<math>\begin{pmatrix}1 & 0 \\ -s & 1 \end{pmatrix} \begin{pmatrix}u & 0 \\ 0 & u \end{pmatrix} \begin{pmatrix}1 & 0 \\ s & 1 \end{pmatrix} = \begin{pmatrix}u & 0 \\ z & u \end{pmatrix}, </math>
where <math>z = u s - s u = \sin \theta (rs - sr) = 2 t \sin \theta .</math>
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== The derivative for quaternions ==
Since the time of Hamilton, it has been realized that requiring the independence of the derivative from the path that a differential follows toward zero is too restrictive: it excludes even <math>f(q) = q^2</math> from differentiability. Therefore a direction-dependent derivative is necessary for functions of a quaternion variable.<ref>
Considering of the increment of polynomial function of quaternionic argument shows that the increment is linear map of increment of the argument.
This statement is the basis for the following definition.
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==See also==
* [[Cayley transform]]
==Notes==
{{notelist}}
==Citations==
{{Reflist|2}}
==References==
* {{Citation
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| first = Vladimir
| author-link = Vladimir Arnold
| translator1-last = Porteous
| translator1-first = Ian R.
| translator-link1 = Ian R. Porteous
| title = The geometry of spherical curves and the algebra of quaternions
| journal = Russian Mathematical Surveys
| volume = 50
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| pages = 1–68
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| doi = 10.1070/RM1995v050n01ABEH001662 | id =
| mr =
| zbl = 0848.58005}}
* {{Citation
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| first = Arthur
| author-link = Arthur Cayley
| title = On the application of quaternions to the theory of rotation
| journal = [[London and Edinburgh Philosophical Magazine]], Series 3
| volume = 33
| issue = 221
| pages = 196–200
| year = 1848
| doi = 10.1080/14786444808645844}}
*{{Citation
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| journal = [[American Mathematical Monthly]]
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| mr =
| zbl = 0282.30040}}
* {{Citation
| last = Du Val
| first = Patrick
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| title = Homographies, Quaternions and Rotations
| place = Oxford
| publisher = Clarendon Press
| series = Oxford Mathematical Monographs
| year = 1964
| mr = 0169108
| zbl = 0128.15403}}
* {{Citation
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| author-link = Rudolf Fueter
| title = Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen
| journal = [[Commentarii Mathematici Helvetici]]
| volume = 8
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* {{Citation
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| publisher = Springer
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* {{Citation
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* {{Citation
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* {{Citation
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* {{Citation
| last = Hamilton
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| author-link = William Rowan Hamilton
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* {{Citation
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* {{Citation
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* {{Citation
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* {{Citation
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[[Category:Quaternions]]
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